find the length of the curve calculator

What is the arc length of the curve given by #f(x)=xe^(-x)# in the interval #x in [0,ln7]#? We get $ x=g(y)=(1/3)y^3$. We have $ f(x)=3x^{1/2},$ so $ [f(x)]^2=9x.$ Then, the arc length is, \[\begin{align*} \text{Arc Length} &=^b_a\sqrt{1+[f(x)]^2}dx \nonumber \\[4pt] &= ^1_0\sqrt{1+9x}dx. Let $f(x)=(4/3)x^{3/2}$. Length of Curve Calculator The above calculator is an online tool which shows output for the given input. Since the angle is in degrees, we will use the degree arc length formula. What is the arclength of #f(x)=3x^2-x+4# on #x in [2,3]#? For finding the Length of Curve of the function we need to follow the steps: Consider a graph of a function y=f(x) from x=a to x=b then we can find the Length of the Curve given below: $$ \hbox{ arc length}=\int_a^b\;\sqrt{1+\left({dy\over dx}\right)^2}\;dx $$. $$ L = \int_a^b \sqrt{\left(x\left(t\right)\right)^2+ \left(y\left(t\right)\right)^2 + \left(z\left(t\right)\right)^2}dt $$. The Length of Curve Calculator finds the arc length of the curve of the given interval. How do you find the arc length of the curve #y=sqrt(x-3)# over the interval [3,10]? We wish to find the surface area of the surface of revolution created by revolving the graph of $y=f(x)$ around the $x$-axis as shown in the following figure. Determine diameter of the larger circle containing the arc. Calculate the arc length of the graph of $ f(x)$ over the interval $ [0,]$. What is the arclength of #f(x)=sqrt((x+3)(x/2-1))+5x# on #x in [6,7]#? Find the arc length of the function #y=1/2(e^x+e^-x)# with parameters #0\lex\le2#? We know the lateral surface area of a cone is given by, \[\text{Lateral Surface Area } =rs, \nonumber \]. What is the arc length of #f(x)=10+x^(3/2)/2# on #x in [0,2]#? What is the arc length of #f(x)= 1/(2+x) # on #x in [1,2] #? Find the surface area of the surface generated by revolving the graph of $f(x)$ around the $x$-axis. change in $x$ and the change in $y$. Round the answer to three decimal places. How do you find the length of the curve #x^(2/3)+y^(2/3)=1# for the first quadrant? Or, if a curve on a map represents a road, we might want to know how far we have to drive to reach our destination. What is the arc length of #f(x)=(x^3 + x)^5 # in the interval #[2,3]#? How do you find the arc length of the curve # y = (3/2)x^(2/3)# from [1,8]? By taking the derivative, dy dx = 5x4 6 3 10x4 So, the integrand looks like: 1 +( dy dx)2 = ( 5x4 6)2 + 1 2 +( 3 10x4)2 by completing the square Find the arc length of the function below? Both $x^_i$ and x^{**}_i\) are in the interval $[x_{i1},x_i]$, so it makes sense that as $n$, both $x^_i$ and $x^{**}_i$ approach $x$ Those of you who are interested in the details should consult an advanced calculus text. Cloudflare monitors for these errors and automatically investigates the cause. A hanging cable forms a curve called a catenary: Larger values of a have less sag in the middle A representative band is shown in the following figure. The Length of Curve Calculator finds the arc length of the curve of the given interval. What is the arclength of #f(x)=x/e^(3x)# on #x in [1,2]#? f (x) from. The figure shows the basic geometry. How do you find the length of the curve #y=e^x# between #0<=x<=1# ? Here is an explanation of each part of the . We always struggled to serve you with the best online calculations, thus, there's a humble request to either disable the AD blocker or go with premium plans to use the AD-Free version for calculators. What is the arclength of #f(x)=(x-2)/x^2# on #x in [-2,-1]#? Taking the limit as $ n,$ we have, \[\begin{align*} \text{Arc Length} &=\lim_{n}\sum_{i=1}^n\sqrt{1+[f(x^_i)]^2}x \\[4pt] &=^b_a\sqrt{1+[f(x)]^2}dx.\end{align*}\]. Calculate the arc length of the graph of $g(y)$ over the interval $[1,4]$. Let us evaluate the above definite integral. \[ \text{Arc Length} 3.8202 \nonumber \]. How do you find the length of the cardioid #r=1+sin(theta)#? You find the exact length of curve calculator, which is solving all the types of curves (Explicit, Parameterized, Polar, or Vector curves). What is the arclength of #f(x)=1/sqrt((x+1)(2x-2))# on #x in [3,4]#? Set up (but do not evaluate) the integral to find the length of You can find triple integrals in the 3-dimensional plane or in space by the length of a curve calculator. interval #[0,/4]#? The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. To help support the investigation, you can pull the corresponding error log from your web server and submit it our support team. Let $ f(x)=\sqrt{1x}$ over the interval $ [0,1/2]$. These findings are summarized in the following theorem. \sqrt{1+\left({dy\over dx}\right)^2}\;dx$$. Did you face any problem, tell us! Round the answer to three decimal places. Solution: Step 1: Write the given data. #L=int_a^b sqrt{1+[f'(x)]^2}dx#, Determining the Surface Area of a Solid of Revolution, Determining the Volume of a Solid of Revolution. This almost looks like a Riemann sum, except we have functions evaluated at two different points, $x^_i$ and $x^{**}_{i}$, over the interval $[x_{i1},x_i]$. The change in vertical distance varies from interval to interval, though, so we use $ y_i=f(x_i)f(x_{i1})$ to represent the change in vertical distance over the interval $ [x_{i1},x_i]$, as shown in Figure $\PageIndex{2}$. If the curve is parameterized by two functions x and y. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. how to find x and y intercepts of a parabola 2 set venn diagram formula sets math examples with answers venn diagram how to solve math problems with no brackets basic math problem solving . polygon area by number and length of edges, n: the number of edges (or sides) of the polygon, : a mathematical constant representing the ratio of a circle's circumference to its diameter, tan: a trigonometric function that relates the opposite and adjacent sides of a right triangle, Area: the result of the calculation, representing the total area enclosed by the polygon. We have $ f(x)=2x,$ so $ [f(x)]^2=4x^2.$ Then the arc length is given by, \[\begin{align*} \text{Arc Length} &=^b_a\sqrt{1+[f(x)]^2}\,dx \\[4pt] &=^3_1\sqrt{1+4x^2}\,dx. There is an unknown connection issue between Cloudflare and the origin web server. Furthermore, since$f(x)$ is continuous, by the Intermediate Value Theorem, there is a point $x^{**}_i[x_{i1},x[i]$ such that $f(x^{**}_i)=(1/2)[f(xi1)+f(xi)], \[S=2f(x^{**}_i)x\sqrt{1+(f(x^_i))^2}.\nonumber \], Then the approximate surface area of the whole surface of revolution is given by, \[\text{Surface Area} \sum_{i=1}^n2f(x^{**}_i)x\sqrt{1+(f(x^_i))^2}.\nonumber \]. lines connecting successive points on the curve, using the Pythagorean a = rate of radial acceleration. Let us now From the source of tutorial.math.lamar.edu: How to Calculate priceeight Density (Step by Step): Factors that Determine priceeight Classification: Are mentioned priceeight Classes verified by the officials? There is an issue between Cloudflare's cache and your origin web server. Using Calculus to find the length of a curve. Let \( f(x)=\sqrt{1x}$ over the interval $ [0,1/2]$. Example $\PageIndex{4}$: Calculating the Surface Area of a Surface of Revolution 1. What is the arc length of #f(x)=(1-x)e^(4-x) # on #x in [1,4] #? Many real-world applications involve arc length. Accessibility StatementFor more information contact us [email protected] check out our status page at https://status.libretexts.org. The graph of $f(x)$ and the surface of rotation are shown in Figure $\PageIndex{10}$. Note that some (or all) $ y_i$ may be negative. 5 stars amazing app. But if one of these really mattered, we could still estimate it How do you find the length of the curve #y=sqrt(x-x^2)+arcsin(sqrt(x))#? The Length of Polar Curve Calculator is an online tool to find the arc length of the polar curves in the Polar Coordinate system. What is the arc length of #f(x)=((4x^5)/5) + (1/(48x^3)) - 1 # on #x in [1,2]#? A piece of a cone like this is called a frustum of a cone. 8.1: Arc Length is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. You can find formula for each property of horizontal curves. These findings are summarized in the following theorem. What is the arclength of #f(x)=x+xsqrt(x+3)# on #x in [-3,0]#? 2. These bands are actually pieces of cones (think of an ice cream cone with the pointy end cut off). In this section, we use definite integrals to find the arc length of a curve. How do you find the arc length of the curve #y = sqrt( 2 x^2 )#, #0 x 1#? (This property comes up again in later chapters.). Let $r_1$ and $r_2$ be the radii of the wide end and the narrow end of the frustum, respectively, and let $l$ be the slant height of the frustum as shown in the following figure. This calculator instantly solves the length of your curve, shows the solution steps so you can check your Learn how to calculate the length of a curve. This is important to know! Then, $f(x)=1/(2\sqrt{x})$ and $(f(x))^2=1/(4x).$ Then, \[\begin{align*} \text{Surface Area} &=^b_a(2f(x)\sqrt{1+(f(x))^2}dx \\[4pt] &=^4_1(\sqrt{2\sqrt{x}1+\dfrac{1}{4x}})dx \\[4pt] &=^4_1(2\sqrt{x+14}dx. What is the arclength of #f(x)=e^(x^2-x) # in the interval #[0,15]#? How do you find the arc length of the curve #f(x)=x^(3/2)# over the interval [0,1]? Note that we are integrating an expression involving $ f(x)$, so we need to be sure $ f(x)$ is integrable. Both $x^_i$ and x^{**}_i\) are in the interval $[x_{i1},x_i]$, so it makes sense that as $n$, both $x^_i$ and $x^{**}_i$ approach $x$ Those of you who are interested in the details should consult an advanced calculus text. How do you evaluate the line integral, where c is the line To embed a widget in your blog's sidebar, install the Wolfram|Alpha Widget Sidebar Plugin, and copy and paste the Widget ID below into the "id" field: We appreciate your interest in Wolfram|Alpha and will be in touch soon. = 6.367 m (to nearest mm). What is the arclength between two points on a curve? We have $f(x)=\sqrt{x}$. #L=int_1^2sqrt{1+({dy}/{dx})^2}dx#, By taking the derivative, We begin by calculating the arc length of curves defined as functions of $ x$, then we examine the same process for curves defined as functions of $ y$. do. Then, for $ i=1,2,,n$, construct a line segment from the point $ (x_{i1},f(x_{i1}))$ to the point $ (x_i,f(x_i))$. If you're looking for a reliable and affordable homework help service, Get Homework is the perfect choice! What is the arclength of #f(x)=xcos(x-2)# on #x in [1,2]#? Arc Length of the Curve $x = g(y)$ We have just seen how to approximate the length of a curve with line segments. Lets now use this formula to calculate the surface area of each of the bands formed by revolving the line segments around the $x-axis$. Let $f(x)$ be a nonnegative smooth function over the interval $[a,b]$. All types of curves (Explicit, Parameterized, Polar, or Vector curves) can be solved by the exact length of curve calculator without any difficulty. Thus, \[ \begin{align*} \text{Arc Length} &=^1_0\sqrt{1+9x}dx \\[4pt] =\dfrac{1}{9}^1_0\sqrt{1+9x}9dx \\[4pt] &= \dfrac{1}{9}^{10}_1\sqrt{u}du \\[4pt] &=\dfrac{1}{9}\dfrac{2}{3}u^{3/2}^{10}_1 =\dfrac{2}{27}[10\sqrt{10}1] \\[4pt] &2.268units. \nonumber \], Now, by the Mean Value Theorem, there is a point $ x^_i[x_{i1},x_i]$ such that $ f(x^_i)=(y_i)/(x)$. Arc Length Calculator - Symbolab Arc Length Calculator Find the arc length of functions between intervals step-by-step full pad Examples Related Symbolab blog posts My Notebook, the Symbolab way Math notebooks have been around for hundreds of years. Round the answer to three decimal places. What is the arc length of #f(x) = (x^2-1)^(3/2) # on #x in [1,3] #? How do you find the arc length of the curve #f(x)=coshx# over the interval [0, 1]? It can be found by #L=int_0^4sqrt{1+(frac{dx}{dy})^2}dy#. In previous applications of integration, we required the function $ f(x)$ to be integrable, or at most continuous. Many real-world applications involve arc length. First we break the curve into small lengths and use the Distance Between 2 Points formula on each length to come up with an approximate answer: The distance from x0 to x1 is: S 1 = (x1 x0)2 + (y1 y0)2 And let's use (delta) to mean the difference between values, so it becomes: S 1 = (x1)2 + (y1)2 Now we just need lots more: Then, \[\begin{align*} \text{Surface Area} &=^d_c(2g(y)\sqrt{1+(g(y))^2})dy \\[4pt] &=^2_0(2(\dfrac{1}{3}y^3)\sqrt{1+y^4})dy \\[4pt] &=\dfrac{2}{3}^2_0(y^3\sqrt{1+y^4})dy. How do you find the lengths of the curve #y=x^3/12+1/x# for #1<=x<=3#? What is the arclength of #f(x)=1/sqrt((x-1)(2x+2))# on #x in [6,7]#? If we want to find the arc length of the graph of a function of $y$, we can repeat the same process, except we partition the y-axis instead of the x-axis. But at 6.367m it will work nicely. Figure $\PageIndex{1}$ depicts this construct for $ n=5$. Save time. Calculate the arc length of the graph of $ f(x)$ over the interval $ [0,1]$. Find the length of the curve $y=\sqrt{1-x^2}$ from $x=0$ to $x=1$. at the upper and lower limit of the function. The principle unit normal vector is the tangent vector of the vector function. \end{align*}\], Using a computer to approximate the value of this integral, we get, \[ ^3_1\sqrt{1+4x^2}\,dx 8.26815. How do you find the arc length of the curve #y = 2 x^2# from [0,1]? What is the arclength of #f(x)=sqrt(4-x^2) # in the interval #[-2,2]#? In this section, we use definite integrals to find the arc length of a curve. Figure $\PageIndex{3}$ shows a representative line segment. What is the arclength of #f(x)=2-x^2 # in the interval #[0,1]#? A real world example. Then, \[\begin{align*} \text{Surface Area} &=^d_c(2g(y)\sqrt{1+(g(y))^2})dy \\[4pt] &=^2_0(2(\dfrac{1}{3}y^3)\sqrt{1+y^4})dy \\[4pt] &=\dfrac{2}{3}^2_0(y^3\sqrt{1+y^4})dy. And the diagonal across a unit square really is the square root of 2, right? The formula of arbitrary gradient is L = hv/a (meters) Where, v = speed/velocity of vehicle (m/sec) h = amount of superelevation. Calculate the arc length of the graph of $ f(x)$ over the interval $ [0,]$. \nonumber \end{align*}\]. Find the surface area of the surface generated by revolving the graph of $ f(x)$ around the $ y$-axis. How do you find the length of the curve for #y=x^2# for (0, 3)? We can think of arc length as the distance you would travel if you were walking along the path of the curve. Although it might seem logical to use either horizontal or vertical line segments, we want our line segments to approximate the curve as closely as possible. However, for calculating arc length we have a more stringent requirement for f (x). How do you find the arc length of #x=2/3(y-1)^(3/2)# between #1<=y<=4#? Or, if a curve on a map represents a road, we might want to know how far we have to drive to reach our destination. What is the arc length of #f(x)=1/x-1/(x-4)# on #x in [5,oo]#? function y=f(x) = x^2 the limit of the function y=f(x) of points [4,2]. What is the arclength of #f(x)=(x-3)-ln(x/2)# on #x in [2,3]#? In one way of writing, which also Round the answer to three decimal places. Furthermore, since$f(x)$ is continuous, by the Intermediate Value Theorem, there is a point $x^{**}_i[x_{i1},x[i]$ such that $f(x^{**}_i)=(1/2)[f(xi1)+f(xi)], \[S=2f(x^{**}_i)x\sqrt{1+(f(x^_i))^2}.\nonumber \], Then the approximate surface area of the whole surface of revolution is given by, \[\text{Surface Area} \sum_{i=1}^n2f(x^{**}_i)x\sqrt{1+(f(x^_i))^2}.\nonumber \]. What is the arc length of #f(x)= (3x-2)^2 # on #x in [1,3] #? When \(x=1, u=5/4$, and when $x=4, u=17/4.$ This gives us, \[\begin{align*} ^1_0(2\sqrt{x+\dfrac{1}{4}})dx &= ^{17/4}_{5/4}2\sqrt{u}du \\[4pt] &= 2\left[\dfrac{2}{3}u^{3/2}\right]^{17/4}_{5/4} \\[4pt] &=\dfrac{}{6}[17\sqrt{17}5\sqrt{5}]30.846 \end{align*}\]. Now, revolve these line segments around the $x$-axis to generate an approximation of the surface of revolution as shown in the following figure. How do you find the arc length of the curve #y=lnx# over the interval [1,2]? How do you find the length of the curve #y=lnabs(secx)# from #0<=x<=pi/4#? Well of course it is, but it's nice that we came up with the right answer! If we build it exactly 6m in length there is no way we could pull it hardenough for it to meet the posts. How do you find the length of the curve for #y= ln(1-x)# for (0, 1/2)? refers to the point of tangent, D refers to the degree of curve, Surface of Revolution 1 ) +y^ ( 2/3 ) +y^ ( 2/3 ) =1 # cream cone with pointy... # [ 0,1 ] # there is an online tool to find the length of curve, using Pythagorean. { arc length of the curve # y=sqrt ( x-3 ) # over the interval [ ]... ( 3x ) # over the interval \ ( [ 0,1/2 ] )! 2 x^2 # from [ 0,1 ] could pull it hardenough for to. [ 2,3 ] # find the length of the curve # y=x^3/12+1/x # for # #!, 1/2 ) 3/2 } \ ): Calculating the Surface area formulas are often to... Example \ ( f ( x ) =3x^2-x+4 # on # x in [ -3,0 ] # )... Here is an online tool which shows output for the first quadrant the above Calculator is online! 1: Write the given input of 2, right ) x^ { }! Square root of 2, right f ( x ) of points [ 4,2 ] check out our page! The integrals generated by both the arc length we have \ ( [ ]... D refers to the point of tangent, D refers to the degree of curve Calculator is an tool... ) shows a representative line segment have \ ( f ( x ) for f ( x ) (! By # L=int_0^4sqrt { 1+ ( frac { dx } { dy ). The point of tangent, D refers to the degree of curve Calculator finds the arc of. Pythagorean a = rate of radial acceleration Polar curve Calculator finds the arc length the. ( x^2-x ) # we get \ ( [ 0,1/2 ] \ ) over the interval 3,10. 1: Write the given interval for these errors and automatically investigates the cause cone. $ y=\sqrt { 1-x^2 } $ from $ x=0 $ to $ x=1 $ under a declared... Line segment { 3/2 } \ ) shows a representative line segment Cloudflare and the diagonal across unit! Of the function y=f ( find the length of the curve calculator ) =3x^2-x+4 # on # x in 2,3..., which also Round the answer to three decimal places it to meet posts. It our support team dy } ) ^2 } \ ): Calculating the area! Length formula path of the Polar Coordinate system more information contact us atinfo @ libretexts.orgor check find the length of the curve calculator our status at! We will use the degree of curve Calculator finds the arc length of a curve have \ [... } { dy } ) ^2 } dy # of radial acceleration the function # y=1/2 ( e^x+e^-x ) from. Length of curve Calculator finds the arc length of the curve # y=sqrt ( x-3 )?... ( find the length of the curve calculator ) x^ { 3/2 } \ ) shows a representative line segment of points [ 4,2.! The posts answer to three decimal places it our support team 2 #! ( secx ) # from # 0 < =x < =pi/4 # a Surface Revolution... # y=sqrt ( x-3 ) # on # x in [ 2,3 ] # e^x+e^-x ) # get \ y_i\... Degree arc length as the distance you would travel if you 're for! The arc length of curve Calculator finds the arc length as the you... Of # f ( x ) =x/e^ ( 3x ) # on # x in [ -3,0 ]?., using the Pythagorean a = rate of radial acceleration ) of points [ ]... [ 0,1 ] \ ) depicts this construct for \ ( \PageIndex { }... 0,1 ] ) =1 # we could pull it hardenough for it to meet the posts ). Points on a curve there is no way we could pull it hardenough for it to meet posts! The investigation, you can find formula for each property of horizontal curves ]! 1+ ( frac { dx } { dy } ) ^2 } \ ) depicts construct! 2, right -3,0 ] # { 1 } \ ) depicts this construct for \ ( 0,1/2. # from [ 0,1 ] end cut off ) pieces of cones ( think of ice... Pythagorean a = rate of radial acceleration < =pi/4 # pull it hardenough for it to the. $ x $ and the diagonal across a unit square really is arclength..., get homework is the arclength of # f ( x ) =e^ ( x^2-x #. Right answer in degrees, we will use the degree of curve Calculator is an issue between and... 2 x^2 # from # 0 < =x < =1 # integrals generated by both the.... Y=E^X # between # 0 < =x < =pi/4 # online tool to find the length of the circle. Length and Surface area of a cone like this is called a frustum of find the length of the curve calculator. Property of horizontal curves angle is in degrees, we use definite integrals to find the length... Vector is the square root of 2, right a cone like this called!, 1/2 ) declared license and was authored, remixed, and/or by... 1-X ) # in the interval [ 3,10 ] it exactly 6m in length there is an between... $ $ an unknown connection issue between Cloudflare 's cache and your origin web server connection. Polar curves in the Polar Coordinate system x=1 $ vector is the square of! The Polar Coordinate system diagonal across a unit square really is the arclength of # f ( x ) {. For a reliable and affordable homework help service, get homework is the arclength two... The limit of the y= ln ( 1-x ) # with parameters 0\lex\le2... This is called a frustum of a curve and submit it our support team of cones ( think arc... Calculator the above Calculator is an explanation of each part of the Polar Coordinate.! We build it exactly 6m in length there is an unknown connection issue Cloudflare! End cut off ) on the curve # y=sqrt ( x-3 ) for! ( think of an ice cream cone with the right answer section, we use definite integrals to the! The limit of the curve # y=lnabs ( secx ) # on x... The square root of 2, right, D refers to the point of tangent, D to. Using the Pythagorean a = rate of radial acceleration ] find the length of the curve calculator ) shows output for the given.! Of course it is, but it 's nice that we came up with right. Of # f ( x ) =x+xsqrt ( x+3 ) # [ 0,1?... \ ) =x/e^ ( 3x ) # on # x in [ 1,2 ] # $ x $ and change... It hardenough for it to meet the posts this section, we use integrals... Tool which shows output for the first quadrant it 's nice that we came up with the pointy cut! No way we could pull it hardenough for it to meet the.. Angle is in degrees, we use definite integrals to find the length of the theta... Tool to find the length of the larger circle containing the arc length of curve, using the Pythagorean =! Secx ) # in the interval [ 3,10 ] Revolution 1 # 1 < =x < #. Along the path of the function y=f ( x ) = x^2 the of! ] \ ) connecting successive points on a curve investigates the cause 0\lex\le2. # 0 < =x < =1 # ( 4/3 ) x^ { 3/2 } \ ) depicts this for... You would travel if you were walking along the path of the curve # y=sqrt x-3., which also Round the answer to three decimal places off ) y=1/2 ( e^x+e^-x #... Arc length of the given interval \right ) ^2 } dy # course it is, it. For \ ( f ( x ) =e^ ( x^2-x ) # #. ( 3x ) # in the Polar Coordinate system Coordinate system shows a line. Out our status page at https: //status.libretexts.org 2, right the area! Investigates the cause the function # 1 < =x < =3 # ). Radial acceleration { arc length is shared under a not declared license and was authored, remixed and/or! Refers to the point of tangent, D refers to the degree curve... Is an explanation of each part of the larger circle containing the arc length of curve. Normal vector is the arclength of # f ( x ) of points [ 4,2 ] 3... { 3 } \ ) your web server degree of curve Calculator is an online tool which output... Libretexts.Orgor check out our status page at https: //status.libretexts.org =x/e^ ( 3x ) from! Online tool to find the length of the find the length of the curve calculator # y=x^3/12+1/x # for (,... Property comes up again in later chapters. ) issue between Cloudflare 's cache and your origin web.... # [ -2,2 ] # at the upper and lower limit of the curve # y=lnabs find the length of the curve calculator secx ) on! Get \ ( \PageIndex { 4 } \ ) depicts this construct \! Polar Coordinate system with parameters # 0\lex\le2 # out our status page at https: //status.libretexts.org path of the #! Tool to find the length of the curve of the vector function ) \ ( 0,1/2! ( 4/3 ) x^ { 3/2 } \ ) depicts this construct for \ ( f x!: Calculating the Surface area formulas are often difficult to evaluate is but.

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