To use this online calculator for Arc Length of Circle given radius and central angle, enter Radius (r) & Central Angle (∠ central) and hit the calculate button. The calculator will then determine the length of the arc. A vector emanating from the zero point can also be used as a pointer. It is not cur Arc Length in Rectangular Coordinates. Of course the traditional way to calculate arc length is to use the formula: L = ∫ a b 1 + ( f ′ ( x)) 2 d x. . The answer is the length of your hypotenuse! The formulas for the arec are: If the angle is alpha, the area is A=pi*r^2*(alpha/360°) and the arc length is b=2*pi*r*(alpha/360). First, let's . First, we will graph our curve and identify the derivative of the function and radius. x d x. The formula for arc length. Finally, the circular segment calculator below includes all possible calculations regarding circular segment parameters: angle. To calculate arc length without radius, you need the central angle and the sector area: Multiply the area by 2 and divide the result by the central angle in radians. Arc Length Formula (s) L = ∫ ds L = ∫ d s where, 2) A circle has an arc length of 5.9 and a central angle of 1.67 radians. Multiply the central angle by the radius to get the arc length. Calculus questions and answers. WonderHowTo. 14, Feb 22. ∫ 14 √1+4 (4)) 2 dx. These integrals often can only be computed using numerical methods. The Arc of a Circle Calculator can also be used to: Find out the radius of a circle, knowing only the diameter. Enter two segment parameters, and the calculator will find all the rest. The length of the curve from to is given by. Many real-world applications involve arc length. Try changing the function! Calculate the Arc Length of a Function of X Question Let the curve C bef (x) = 9x + 1 over the interval 5 x < 8. This is now we are integrating a bunch of dx's or we're integrating with respect to x. 8. Find the length of an arc, using the chord length and arc angle. Use this information to determine the position after traveling ˇ p 10 units. However you choose to think about calculating arc length, you will get the formula L = Z 5 5 p How do you find the length of the curve y = x5 6 + 1 10x3 between 1 ≤ x ≤ 2 ? Example 2 Determine the arc length function for →r (t) = 2t,3sin(2t),3cos(2t) r → ( t) = 2 t, 3 sin ( 2 t), 3 cos ( 2 t) . The ArcLength(f(x), x=a..b) command returns the arc length of the expression expression f ⁡ x from a to b. We define the arc length function as, s(t) = ∫ t 0 ∥∥→r ′(u)∥∥ du s ( t) = ∫ 0 t ‖ r → ′ ( u) ‖ d u Before we look at why this might be important let's work a quick example. It is gotten by just calculating a certain angle of a circle. Added Mar 1, 2014 by Sravan75 in Mathematics. Free Arc Length calculator - Find the arc length of functions between intervals step-by-step. Click the "Arc Length" button, input radius 3.6 then click the "DEGREES" button. 4.3.1 Examples Example 4.3.1.1 Find the length of the curve ~ r (t)=h3cos(t),3sin(t),ti when 5 t 5. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step Taking a limit then gives us the definite integral formula. Similarly, if x = g ( y) with g continuously differentiable on [ c, d], then the arc length L of g ( y) over [ c, d] is given by. Solution: Given. 32/3 + 8. So, the integrand looks like: √1 +( dy dx)2 = √( 5x4 6)2 + 1 2 +( 3 10x4)2. by . The goal with calculating the arc length of a function is to get as close as possible to the actual length of the graph from starting point a, to ending point b. Does anyone know of any analytical methods to anti-differentiate 1 + cos 2. Question: Find the arc length of the graph of the function over the indicated interval. Here is how the Arc Length of Circle given radius and central angle calculation can be explained with given input values -> 7.853982 = 10*0.785398163397301 . According to this formula arc length of a circle is equals to: The central angle θ in radians. 2.4.2 Determine the length of a curve, between two points. The curvature of a circle is equal to the reciprocal of its radius. Share a link to this widget: More. Arc Length - Non-Calculator.notebook 4 June 29, 2017 May 10-15:10. 4. b = 2. ∫ 0 3 π 4 cos 2 ( 2 t ) + sin 2 ( t ) + 1 d t . Add; Subtract; Multiply; Divide; Multiple Operations; Prime Factorization; Elementary Math. Let's first begin by finding a general formula for computing arc length. Click the "Radius" button, input arc length 5.9 and central angle 1.67. This means that we can calculate the length of the vector, $\textbf {u} = \left<2, 4, -1\right>$, by . Math24.pro Math24.pro. Show Solution This video discusses the formula for finding arc length if a curve is given in parametric form. It is relatively straight forward to calculate the arc length of a catenary described by the function y = a cosh (x/a). Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called rectification of a curve. Let a curve C be defined by the equation y = f (x) where f is continuous on an interval [a, b]. We could say, "Okay, x equals a to x equals b." Let's take the sum of the product of this expression and dx, and this is essential. ; 3.3.3 Describe the meaning of the normal and binormal vectors of a curve in space. - To estimate the arc length of the function, we can add the length of the segments from y1 to y10. An arc is a part of a circle. Provide your answer below: Calculate the Surface Area of a Surface of Revolution Around the Y-Axis Question Consider the curve f (y . Plotters draw lines on paper using a pen, or in some applications, use a knife to cut a material like vinyl or leather.In the latter case, they are sometimes known as a cutting plotter.. . They are just di↵erent ways of writing the same thing. Arc Length Calculator The arc automatically calculates when you input the size and angle. Arc Length of Polar Curve. (Please read about Derivatives and Integrals first) Imagine we want to find the length of a curve between two points. A vector emanating from the zero point can also be used as a pointer. We have, Sector area = 25 units. Arc Length Formula. This arc length problem involves non circular arcs, it is talking about "curve length", i.e. 2022 Math24.pro info@math24.pro info@math24.pro We recall that if f is a smooth curve and f ′ is continuous on the closed interval [a,b], then the length of the curve is found by the following Arc Length Formula: L = ∫ a b 1 + ( f ′ ( x)) 2 d x Arc Length Of A Parametric Curve We can think of arc length as the distance you would travel if you were walking along the path of the curve. The ArcLength([f(x), g(x)], x=a..b) command returns the parametric arc length expressed in cartesian coordinates. It is an online Geometry tool requires radius length of a circle. To find the arc length of the vector function, we will need to use the formula. S A = ∫ − 1 1 2 π ( 4 − x 2) ⏟ radius 1 + ( − x 4 − x 2) 2 ⏟ length d x S A . L = ∫ c d 1 + [ g ′ ( y)] 2 d y. Lets use the above formula to calculate the arc length of circle. This video gives the formula to find the arc length of a 3-dimensional vector function and do one concrete example of finding the length of a vector function. Send feedback | Visit Wolfram|Alpha. If the rectification of a curve results in a finite number (so the curve has a finite length), then the curve is said to be rectifiable. By using this website, you agree to our Cookie Policy. Problem 3: Calculate the arc length of a curve sector area 25cm 2 and radius . Arc length Cartesian Coordinates. This is the formula for arc length. We begin by defining a function f(x), like in the graph below. 01, Mar 22 . Take the square root of the sum to return the length of the vector. It's true! From this point on we are going to use the following formula for the length of the curve. COMPUTING THE ARC LENGTH OF A DIFFERENTIABLE FUNCTION ON A CLOSED INTERVAL . 5. s=. To find the length of the curve between x = x o and x = x n, we'll break the curve up into n small line segments, for which it's easy to find the length just using the Pythagorean theorem, the basis of how we calculate distance on the plane.. [8, 125] The figure shows the basic geometry. Generally the arc length, s, of a real function f (x), with a derivative f' (x), that is continuous on [-b,b] can be found as s = Integrate [Srqt [1+f' (x)^2], {x,-b,b}] Check if f (x) = x2 +2x f ( x) = x 2 + 2 x is continuous. Length Contraction Formula. Get 10% discount. See also. Enter central angle =63.8 then click "CALCULATE" and your answer is Arc Length = 4.0087. In this case, there is no real number that makes the expression undefined. This conversion would give the accurrate answers: 180 degrees * 30 pi / 360 = 15 pi. It does not extend because these arcs don't have circular properties. Square that, too. The norm of a vector is also called the length of a vector. In this tutorial, we learn how to find the arc length of a function. Arc Length - Non-Calculator.notebook 1 June 29, 2017 Oct 26-07:57 Arc Length and Sector Area - Lesson 2. Example 1.0.0.1 Reparametrize ~r(t) = h2t;3sin(2t);3cos(2t)i by its arc length starting from (0;0;3). The graph below . Answer (1 of 3): This is pretty advanced calculus (at least it was for me the first time I tried it). The principal unit normal vector at t is defined to be. Step 2: 50/radius 2 = 50/4 = 12.5 = central angle (rad) There are two basic methods or formulas to calculate the length of the chord. The same process can be applied to functions of ; The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution. Arc length formula is used to calculate the measure of the distance along the curved line making up the arc (a segment of a circle). If an input is given then it can easily show the result for the given number. Problem 2: Find the arc length of a curve on a circle with a radius of 16cm and a central angle of 4radians. By taking the derivative, dy dx = 5x4 6 − 3 10x4. What is an arc? The formula to measure the length of the arc is -. 6. l = 1 0 ∑ n = 1 f a + nc − f a + n − 1 c 2 + c 2. the equation for the upper half of the ellipse can be written as a function y = f(x). 05, Feb 22. The formula says if your function is in 'y=x' you have to take one times the . It is not cur The parameter \(s\) is the arc length parameter if, and only if, \(\norm{\vrp(s)} = 1\text{. The figure shows a graph of the function : In order to find out the arc length of the curve of the line shown in the figure, one should calculate the definite integral: More generally, if we have a function defined in Cartesian coordinates and the task is to find the arc length of the curve between the . Arc Length - Non-Calculator.notebook 3 June 29, 2017 May 10-14:51. The arc length is first approximated using line segments, which generates a Riemann sum. Length of arc formula = θ × 2 A ÷ θ. Imagine this as a string you're pulling straight to see how long it is going to be. Add the squares of these components. radius. Arc length Cartesian Coordinates If we use Leibniz notation for derivatives, the arc length is expressed by the formula. Let us look at some details. 10/30/10 2:57 PM. we may write this: (1) d r = α r d θ. which gives, for the element of length d l : d l = ( d r 2 + ( r d θ) 2)) 1 2 = β d r. Embed this widget ». The arc length formula says the length of the curve is the integral of the norm of the derivatives of the parameterized equations. The binormal vector at t is defined as where is the unit tangent vector. By using this website, you agree to our Cookie Policy. Rest of Steps. Calculus: Integral with adjustable bounds. By Cross multiply: 12sin1000 = asin500 12 s i n 100 0 = a s i n 50 0. In fact, the following theorem states that this characterizes the arc length parameter. 8. f (x) = x2 + 2x f ( x) = x 2 + 2 x , 0 < x < 6 0 < x < 6. Central angle = 2 units. example. We will assume that the derivative f '(x) is also continuous on [a, b]. By using this website, you agree to our Cookie Policy. One application of a definite integral is to calculate the arc length of a plane curve. One application of a definite integral is to calculate the arc length of a plane curve. Insert f(x), which in this case is: 3x^(3/2) There are several different formulas for curvature. The radius of the circle is 16cm. Arc Length of 2D Parametric Curve. Arc Length (Calculus) Arc Length Using Calculus to find the length of a curve. Arc Length of Polar Curve Calculator - Math24.pro Free Arc Length of Polar Curve calculator - Find the arc length of functions between intervals step-by-step Math24.proMath24.pro Arithmetic Add Subtract Multiply Divide Multiple Operations Prime Factorization Elementary Math Simplification Expansion Factorization Completing the Square Calculus. 20. powered by . 3.3.1 Determine the length of a particle's path in space by using the arc-length function. Derivative of Function Calculator. If we are only given the diameter and not the radius we can enter that instead, though the radius is always half the diameter so it's not too difficult to calculate. Figure 1. The key to take from it is that \(\vrp(s)\) is a unit vector. Arc Length Lesson Lesson Contents What is Arc Length? The formula to measure the length of the arc is -. 7. Plotters draw lines on paper using a pen, or in some applications, use a knife to cut a material like vinyl or leather.In the latter case, they are sometimes known as a cutting plotter.. . From a to b is the range of the funtion. Estimate: Looking at the graph (which has equal scaling along the x - and y -axes), we can see the final answer should be a little more than 10 m, somewhere between 10 m and 11 m. Now to find the exact length: `text [length]=r=int_a^b sqrt [1+ ( (dy)/ (dx))^2]\ dx`. The above calculator is an online tool which shows output for the given input. We can use this function to reparametrize ~r(t) so that our input is the distance we traveled from t= a. This calculator, makes calculations very simple and interesting. . This video provides an example of how to determine the arc length of a quadratic function using integration. area. Verify the result using the arc length calculator. L = ∫ a b 1 + [ f ′ ( x)] 2 d x. How to calculate the arc length of a vector function. Next, we substitute everything into our surface area formula and integrate! Inputs the parametric equations of a curve, and outputs the length of the curve. Solution : Distance of chord from center of the circle = 15 cm. the characteristic feature of a logarithmic (or 'equi-angular') spiral is central self-similarity, expressed geometrically as a proportionality between the two elements of length in polar cordinates. The figure shows a graph of the function : In order to find out the arc length of the curve of the line shown in the figure, one should calculate the definite integral: More generally, if we have a function defined in Cartesian coordinates and the task is to find the arc length of the curve between the . And so obviously the formula becomes in this case: ∫ 0 2 π 1 + cos 2. Square that, too. By Cross multiply: 12sin1000 = asin500 12 s i n 100 0 = a s i n 50 0. L = 0.37 units. This program TI-83 Plus and TI-84 Plus program calculates arc length and surface area of a function. a chord length can be determined by using the perpendicular distance from the centre of the circle as well as by the trigonometric method. First zoom in, or enter the address of your starting point. 1. f x = x 2 . Laplace Transform Calculator. Therefore, 0 is also a complex number and can be represented as 0+0i. But I am stuck here. As my example, I will go ahead and randomly choose the quadratic y = 2x^2 - 5x + 3 and find its arc length for x from 1 to 4 To find the length of any function y = f(x) from x= a to x=b we will. Answer. Calculus: Fundamental Theorem of Calculus Estimate the diameter of a circle when its radius is known. Compute the arc angle by inserting the values of the arc length and radius. Here's a breakdown of the steps to calculate the vector's length: List down the components of the vector then take their squares. Try the free Mathway calculator and problem solver below to practice various math topics. What is the radius? ; 3.3.2 Explain the meaning of the curvature of a curve in space and state its formula. Square root of 2 times the area A that is divided by θ. The following problems involve the computation of arc length of differentiable functions on closed intervals. To clarify, let's look at some examples. Because these to square roots do not simplify any further, you have to express the arc length for this function and these bounds as √257 - √17. arc length = (central angle x π/180 ) x radius. In this case, you get the square root of 257. height. Tap for more steps. The answer is the length of your hypotenuse! Solution: As the central angle is given in radians we use the formula, L = θ × (π/180) × r. L = 6 × (3.14 / 180) × 3.5. The units will be the square root of the sector area units. Now, Arc length=θ×r =>4×16 =>64cm. The domain of the expression is all real numbers except where the expression is undefined. 15 circular segment calculations in one program. 2.4.3 Find the surface area of a solid of revolution. Formula for finding arc length of a vector function. Find the arc length of the curve C Give an exact answer. The major difference between arc length and sector area is that an arc is a part of a curve whereas A sector is part of a circle that is enclosed . The central angle(θ) is 4 radians. Hence, the arc length is 8cm. Then the arc length L of f ( x) over [ a, b] is given by. We can find the arc length to be 1261 240 by the integral. Free Arc Length calculator - Find the arc length of functions between intervals step-by-step This website uses cookies to ensure you get the best experience. Surface Area . L = ∫ 2 1 √1 + ( dy dx)2 dx. Size = Angle = Arc length = 62.8319 You have one free use of this calculator. 10% Off Your First Order; Arithmetic. Outputs the arc length and graph. Arc Length from a to b = Z b a |~ r 0(t)| dt These equations aren't mathematically di↵erent. In simple words, the distance that runs through the curved line of the circle making up the arc is known as the arc length. }\) Subsection 11 . not a straight line (though equally aplicable if you wished). Approximate Length. Example: Calculate the arc length of a curve with sector area 25 square units and radius as 2 units. arc length = (25 x π/180 ) x 3. arc length = (25 x π/180 ) x 3. arc length = (0.43633231299 ) x 3. arc length = 1.308996939 m. Example 2 : Find arc length of a wooden wheel with diameter measuring 3 ft and central angle of 45 . And we wish to find the area of the surface obtained by rotating this arc about the x-axis. Now, use the arc length calculator to find the arc length with the given dimensions: Radius = 20 units, Angle = 2 radians Putting the two numbers beneath the same radical will be incorrect; √9-4 is . . 3. a = − 2. Note: Set z (t) = 0 if the curve is only 2 dimensional. Arc Length Formula: A continuous part of a curve or a circle's circumference is called an arc.The curved portion of all objects is mathematically called an arc.If two points are chosen on a circle, they divide the circle into one major arc and one minor arc or two semi-circles. Free Arc Length calculator - Find the arc length of functions between intervals step-by-step This website uses cookies to ensure you get the best experience. Step 1: Sector area × 2 = 25 × 2 = 50. Define the integrand as an anonymous function. . How to Calculate the Length of a Curve The formula for calculating the length of a curve is given as: L = ∫ a b 1 + ( d y d x) 2 d x Where L is the length of the function y = f (x) on the x interval [a, b] and is the derivative of the function y = f (x) with respect to x. The result for the given input the binormal vector at t is defined to be 1261 240 by the.... 4 ) ) 2 dx have a part of the expression is undefined: Set z ( )! Values of the circle = 50 into our surface area of an,. ; 3.3.2 Explain the meaning of the normal and binormal vectors of a circle is equal to the of! X2 +2x f ( x ) easily show the result for the length of an arc arc length of a function calculator... ′ ( y = f ( x ) is 4 radians diameter of a curve space. The curve is smooth ( the derivative of the expression is all real numbers except where the is... ) x radius curvature of a circle along the path of the curve get the length... June 29, 2017 May 15-13:57 your function is in & # x27 ; ( & # 92 (! Of arc length of the circle as well as by the central angle ( θ ) is also a number. In this tutorial, we learn how to find the arc length of 5.9 and central angle again to the... A general formula for finding arc length - Non-Calculator.notebook 4 June 29 2017! > the above calculator is an online Geometry tool requires radius length of the arc length of a vector from! Is only 2 dimensional 6 − 3 10x4 is equal to the reciprocal of its radius its formula following involve. Arc-Length function root by the integral area formula and integrate long it is talking about quot!, 2017 May 10-14:51 take one times the area a that is by... Mathway calculator and problem solver below to practice various Math topics or y. Inputs the equation intervals... Unit tangent vector by Sravan75 in Mathematics by Cross multiply: 12sin1000 = asin500 12 s i n 100 =... Arc length=θ×r = & gt ; 64cm 4 ) ) 2 dx anti-differentiate 1 + 2... Length and radius this as a pointer along a curve, and the... ( t ) + sin 2 ( 2 t ) + 1 d t use Leibniz notation Derivatives... ) x radius revolution Around the Y-Axis Question Consider the curve is given in parametric.... 25 × 2 = 25 × 2 = 50 times the ∫ 1... Not extend because these arcs don & # 92 ; vec r ( )! The circular segment - PLANETCALC < /a > the above calculator is online! Root of 2 times the area a that is divided by θ: the angle. Lesson Lesson Contents What is arc length of the ellipse can be represented as 0+0i of the segments from to. Sin 2 ( 2 t ) + sin 2 ( 2 t ) = 0 if the curve to! A href= '' https: //www.examhill.com/length-curve-calculator/ '' > 11 part of the vector function, we how... Geometry tool requires radius length of a particle & # x27 ; &... Divide ; Multiple Operations ; Prime Factorization ; Elementary Math, let & # 92 ; r. Check if f ( x ) is also a complex number and can be written as a string &. Re pulling straight to See how long it is an online tool which shows output for the given input is! Number that makes the expression is undefined divided by θ traveling ˇ p 10.... Integral formula n 100 0 = a s i n 50 0 answer to three decimal.. N 100 0 = a s i n 100 0 = a s i 50! Curve, and the curve by finding a general formula for computing arc length - Non-Calculator.notebook 2 29! Point can also be used as a pointer the arc-length function tutorial, we use Leibniz notation for,. Case: ∫ 0 2 π 1 + cos 2 ( 2 t ) + d... ; re pulling straight to See how long it is going to be 1261 240 by the.... Circle has an arc using the arc-length function non circular arcs, it is gotten by calculating! Formula and integrate vectors of a curve below to practice various Math topics and central angle again to get arc... Try the free Mathway calculator and problem solver below to practice various topics. ( dy dx = 5x4 6 − 3 10x4 > calculus - arc length a. Enter central angle ( θ ) is also continuous on [ a b! Methods to anti-differentiate 1 + [ g ′ ( x ) ] 2 d x length the! //Planetcalc.Com/1421/ '' > online calculator: circular segment - PLANETCALC < /a > the above calculator is an online tool! Of curve calculator - www.examhill.com < /a > the above calculator is online... The curvature of a circle on [ a, b ] arc, the... And radius as 2 units only 2 dimensional clarify, let & # x27 (... One point to another point along a curve using numerical methods a plot or inert integral.! You wished ) Imagine this as a function y = f ( x.! The perpendicular distance from the zero point can also be used as a function y = f ( )! We learn how to find the arc length = 4.0087 input into four:... Angle θ in radians will be incorrect ; √9-4 is ; Calculate & quot button! A curve is smooth ( the derivative of the curve a solid of revolution anyone know of analytical. Re pulling straight to See how long it is an online tool which output... Given in parametric form 19, 2016 by Sravan75 in Mathematics have to take one times the distance you travel! S first begin by finding a general formula for finding arc length is expressed the! Curve in space number and can be represented as 0+0i is - along a curve line ( though equally if! Tutorial, we will assume that the command returns a plot or inert integral instead which. To use the following problems involve the computation of arc length parameter,... This section, we use definite integrals to find the arc is.! The circle as well as by the integral is known root of the area... ; re pulling straight to See how long it is an online Geometry tool radius. Traveled from one point to another point along a curve in space and its! Three decimal places. 2 + 2 x is continuous ) to clarify, let & # x27 ; &... Length & quot ;, i.e for arcs are similar to circle formulas but! = 5x4 6 − 3 10x4 real numbers except where the expression is undefined equations of circle... With sector area 25cm 2 and radius as 2 units is divided by θ for,. ( central angle again to get the square root of 257, the following states... A complex number and can be represented as 0+0i practice various Math topics segment PLANETCALC! Www.Examhill.Com < /a > the above calculator is an online tool which shows output for the of... Have one free use of this calculator show the result for the given input Derivatives... We are going to use the formula arc length of a function calculator if your function is in & # 92 ; vec (. ( though equally aplicable if you wished ), 2017 May 10-14:51 a central angle x ). Π 1 + cos 2 ( 2 t ) + sin 2 ( t +... The range of the circle = 15 cm the chord length and.... Same radical will be incorrect ; √9-4 is 3.3.3 Describe the meaning of the curve get the square root the... Is also a complex number and can be determined by using this website, can... The equation and intervals to compute the range of the curve: ∫ 0 3 4! Arc using the chord length and radius arc, using the arc length of the is. − 3 10x4 angle θ in radians at t is defined to be 1261 240 by the formula for given. Sector calculator - www.examhill.com < /a > the above calculator is an online Geometry tool requires radius of... Then it can easily show the result for the upper half of curve! We can find the arc length of the sector area units sector calculator - High calculation... [ f ′ ( x ) ] 2 d x Multiple Operations ; Prime Factorization ; Math! Derivative, dy dx ) 2 dx Derivatives and integrals first ) Imagine we to! Think of arc length of the arc length formula in terms of or! And binormal vectors of a curve, and the curve f ( x ) = 0 the... Function and radius as 2 units formula and integrate numbers beneath the same radical will be the square of! To See how long it is talking about & quot ; curve length & quot curve... Can specify that the derivative, dy dx = 5x4 6 − 3 10x4 above calculator is an online tool. Answer below: Calculate the surface area of an arc, using the function... Differentiable functions on closed intervals Operations ; Prime Factorization ; Elementary Math extend these... Can also be used as a string you & # x27 ; s look at some examples of... Www.Examhill.Com < /a > See also: circular segment parameters: angle similar to circle formulas, but mind you. Into our surface area formula and integrate ; √9-4 is how long it is gotten just. As where is the range of the funtion problems involve the computation of length. All real numbers except where the expression undefined answer to three decimal places )!
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