The formula for calculating the area of a regular polygon (a polygon with all sides and angles equal) given the number of edges (n) and the length of one edge (s) is: Area = (n x s) / (4 x tan (/n)) where is the mathematical constant pi (approximately 3.14159), and tan is the tangent function. R = 5729.58 / D T = R * tan (A/2) L = 100 * (A/D) LC = 2 * R *sin (A/2) E = R ( (1/ (cos (A/2))) - 1)) PC = PI - T PT = PC + L M = R (1 - cos (A/2)) Where, P.C. Then, the surface area of the surface of revolution formed by revolving the graph of \(f(x)\) around the x-axis is given by, \[\text{Surface Area}=^b_a(2f(x)\sqrt{1+(f(x))^2})dx \nonumber \], Similarly, let \(g(y)\) be a nonnegative smooth function over the interval \([c,d]\). Calculate the arc length of the graph of \( f(x)\) over the interval \( [0,1]\). \[ \dfrac{}{6}(5\sqrt{5}3\sqrt{3})3.133 \nonumber \]. Length of Curve Calculator The above calculator is an online tool which shows output for the given input. We can then approximate the curve by a series of straight lines connecting the points. 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. To gather more details, go through the following video tutorial. By the Pythagorean theorem, the length of the line segment is, \[ x\sqrt{1+((y_i)/(x))^2}. Lets now use this formula to calculate the surface area of each of the bands formed by revolving the line segments around the \(x-axis\). Calculate the arc length of the graph of \( f(x)\) over the interval \( [1,3]\). How do you find the arc length of the curve #y= ln(sin(x)+2)# over the interval [1,5]? What is the arc length of #f(x)=(1-x)e^(4-x) # on #x in [1,4] #? Find the surface area of the surface generated by revolving the graph of \( f(x)\) around the \( y\)-axis. How do you find the arc length of the curve # y = (3/2)x^(2/3)# from [1,8]? Polar Equation r =. Taking a limit then gives us the definite integral formula. What is the arclength of #f(x)=e^(1/x)/x# on #x in [1,2]#? What is the arclength of #f(x)=x-sqrt(x+3)# on #x in [1,3]#? More. These bands are actually pieces of cones (think of an ice cream cone with the pointy end cut off). In previous applications of integration, we required the function \( f(x)\) to be integrable, or at most continuous. Added Mar 7, 2012 by seanrk1994 in Mathematics. \nonumber \end{align*}\]. What is the arc length of #f(x) = -cscx # on #x in [pi/12,(pi)/8] #? What is the arclength of #f(x)=(x^2+24x+1)/x^2 # in the interval #[1,3]#? 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? Solving math problems can be a fun and rewarding experience. What is the arclength of #f(x)=ln(x+3)# on #x in [2,3]#? What is the arclength of #f(x)=3x^2-x+4# on #x in [2,3]#? Find the length of the curve to. What is the arc length of #f(x)=secx*tanx # in the interval #[0,pi/4]#? 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 . How do you find the arc length of the curve #y=lnx# from [1,5]? }=\int_a^b\; How do you find the lengths of the curve #y=x^3/12+1/x# for #1<=x<=3#? Performance & security by Cloudflare. It may be necessary to use a computer or calculator to approximate the values of the integrals. What is the arclength of #f(x)=sqrt(x^2-1)/x# on #x in [-2,-1]#? Let \( f(x)\) be a smooth function over the interval \([a,b]\). Determine the length of a curve, \(y=f(x)\), between two points. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The integral is evaluated, and that answer is, solving linear equations using substitution calculator, what do you call an alligator that sneaks up and bites you from behind. find the length of the curve r(t) calculator. Figure \(\PageIndex{1}\) depicts this construct for \( n=5\). How do you find the length of the curve #y=lnabs(secx)# from #0<=x<=pi/4#? For \(i=0,1,2,,n\), let \(P={x_i}\) be a regular partition of \([a,b]\). You write down problems, solutions and notes to go back. How do you find the length of the curve for #y=x^(3/2) # for (0,6)? Dont forget to change the limits of integration. Arc Length Calculator. Calculate the arc length of the graph of \( f(x)\) over the interval \( [0,1]\). Use the process from the previous example. What is the arclength of #f(x)=(x-2)/(x^2+3)# on #x in [-1,0]#? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. You can find the double integral in the x,y plane pr in the cartesian plane. It can be quite handy to find a length of polar curve calculator to make the measurement easy and fast. As we have done many times before, we are going to partition the interval \([a,b]\) and approximate the surface area by calculating the surface area of simpler shapes. \[ \dfrac{1}{6}(5\sqrt{5}1)1.697 \nonumber \]. segment from (0,8,4) to (6,7,7)? Then, the surface area of the surface of revolution formed by revolving the graph of \(g(y)\) around the \(y-axis\) is given by, \[\text{Surface Area}=^d_c(2g(y)\sqrt{1+(g(y))^2}dy \nonumber \]. How do you find the distance travelled from t=0 to t=3 by a particle whose motion is given by the parametric equations #x=5t^2, y=t^3#? Calculate the arc length of the graph of \( f(x)\) over the interval \( [0,]\). The basic point here is a formula obtained by using the ideas of calculus: the length of the graph of y = f ( x) from x = a to x = b is arc length = a b 1 + ( d y d x) 2 d x Or, if the curve is parametrized in the form x = f ( t) y = g ( t) with the parameter t going from a to b, then arc length = a b ( d x d t) 2 + ( d y d t) 2 d t 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 arclength of #f(x)=[4x^22ln(x)] /8# in the interval #[1,e^3]#? Legal. However, for calculating arc length we have a more stringent requirement for f (x). How do you find the distance travelled from t=0 to #t=2pi# by an object whose motion is #x=cos^2t, y=sin^2t#? How do you find the length of a curve in calculus? Here is an explanation of each part of the formula: To use this formula, simply plug in the values of n and s and solve the equation to find the area of the regular polygon. How do you find the length of the line #x=At+B, y=Ct+D, a<=t<=b#? I love that it's not just giving answers but the steps as well, but if you can please add some animations, cannot reccomend enough this app is fantastic. How do you find the arc length of the curve #y=2sinx# over the interval [0,2pi]? Conic Sections: Parabola and Focus. Then the formula for the length of the Curve of parameterized function is given below: $$ \hbox{ arc length}=\int_a^b\;\sqrt{\left({dx\over dt}\right)^2+\left({dy\over dt}\right)^2}\;dt $$, It is necessary to find exact arc length of curve calculator to compute the length of a curve in 2-dimensional and 3-dimensional plan, Consider a polar function r=r(t), the limit of the t from the limit a to b, $$ L = \int_a^b \sqrt{\left(r\left(t\right)\right)^2+ \left(r\left(t\right)\right)^2}dt $$. A piece of a cone like this is called a frustum of a cone. Let \(f(x)=\sqrt{x}\) over the interval \([1,4]\). In mathematics, the polar coordinate system is a two-dimensional coordinate system and has a reference point. Find the surface area of a solid of revolution. Our team of teachers is here to help you with whatever you need. Taking the limit as \( n,\) we have, \[\begin{align*} \text{Arc Length} &=\lim_{n}\sum_{i=1}^n\sqrt{1+[f(x^_i)]^2}x \\[4pt] &=^b_a\sqrt{1+[f(x)]^2}dx.\end{align*}\]. \nonumber \]. To embed this widget in a post on your WordPress blog, copy and paste the shortcode below into the HTML source: To add a widget to a MediaWiki site, the wiki must have the. Let \(f(x)\) be a nonnegative smooth function over the interval \([a,b]\). The arc length of a curve can be calculated using a definite integral. We define the arc length function as, s(t) = t 0 r (u) du s ( t) = 0 t r ( u) d u. Embed this widget . All types of curves (Explicit, Parameterized, Polar, or Vector curves) can be solved by the exact length of curve calculator without any difficulty. (This property comes up again in later chapters.). $\begingroup$ @theonlygusti - That "derivative of volume = area" (or for 2D, "derivative of area = perimeter") trick only works for highly regular shapes. provides a good heuristic for remembering the formula, if a small We have \(f(x)=\sqrt{x}\). The formula for calculating the area of a regular polygon (a polygon with all sides and angles equal) given the number of edges (n) and the length of one edge (s) is: where is the mathematical constant pi (approximately 3.14159), and tan is the tangent function. In this section, we use definite integrals to find the arc length of a curve. To help support the investigation, you can pull the corresponding error log from your web server and submit it our support team. If we build it exactly 6m in length there is no way we could pull it hardenough for it to meet the posts. 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 #sqrt(4-x^2)# from [-2,2]? example \nonumber \]. Functions like this, which have continuous derivatives, are called smooth. We can think of arc length as the distance you would travel if you were walking along the path of the curve. Round the answer to three decimal places. What is the arc length of #f(x) = x-xe^(x) # on #x in [ 2,4] #? What is the arc length of #f(x) = (x^2-1)^(3/2) # on #x in [1,3] #? How do you find the length of the curve #y=x^5/6+1/(10x^3)# between #1<=x<=2# ? Let \( g(y)=\sqrt{9y^2}\) over the interval \( y[0,2]\). How do I find the arc length of the curve #y=ln(sec x)# from #(0,0)# to #(pi/ 4, ln(2)/2)#? How do you find the arc length of the curve #f(x)=x^3/6+1/(2x)# over the interval [1,3]? #sqrt{1+({dy}/{dx})^2}=sqrt{({5x^4)/6)^2+1/2+(3/{10x^4})^2# by cleaning up a bit, = cos2( 3)sin( 3) Let us first look at the curve r = cos3( 3), which looks like this: Note that goes from 0 to 3 to complete the loop once. Find the arc length of the curve along the interval #0\lex\le1#. It can be found by #L=int_0^4sqrt{1+(frac{dx}{dy})^2}dy#. The formula for calculating the length of a curve is given below: $$ \begin{align} L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \: dx \end{align} $$. How do you find the length of a curve defined parametrically? How do you find the arc length of the curve #y=(5sqrt7)/3x^(3/2)-9# over the interval [0,5]? What is the arc length of #f(x)=1/x-1/(x-4)# on #x in [5,oo]#? The formula for calculating the length of a curve is given below: L = a b 1 + ( d y d x) 2 d x How to Find the Length of the Curve? The arc length is first approximated using line segments, which generates a Riemann sum. If you're looking for support from expert teachers, you've come to the right place. TL;DR (Too Long; Didn't Read) Remember that pi equals 3.14. How do you set up an integral for the length of the curve #y=sqrtx, 1<=x<=2#? How do you find the length of the curve for #y=2x^(3/2)# for (0, 4)? Everybody needs a calculator at some point, get the ease of calculating anything from the source of calculator-online.net. Let \( f(x)\) be a smooth function defined over \( [a,b]\). How do you find the arc length of the curve #y = 4x^(3/2) - 1# from [4,9]? \[ \begin{align*} \text{Surface Area} &=\lim_{n}\sum_{i=1}n^2f(x^{**}_i)x\sqrt{1+(f(x^_i))^2} \\[4pt] &=^b_a(2f(x)\sqrt{1+(f(x))^2}) \end{align*}\]. #L=int_1^2sqrt{1+({dy}/{dx})^2}dx#, By taking the derivative, How do you find the length of the curve #y=3x-2, 0<=x<=4#? What is the arc length of #f(x)=2-3x# on #x in [-2,1]#? The graph of \( g(y)\) and the surface of rotation are shown in the following figure. Many real-world applications involve arc length. Note that we are integrating an expression involving \( f(x)\), so we need to be sure \( f(x)\) is integrable. \nonumber \]. 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Determine the length of a curve, \(x=g(y)\), between two points. How do you find the length of the curve #y=sqrt(x-x^2)#? This calculator, makes calculations very simple and interesting. refers to the point of tangent, D refers to the degree of curve, You just stick to the given steps, then find exact length of curve calculator measures the precise result. Although it is nice to have a formula for calculating arc length, this particular theorem can generate expressions that are difficult to integrate. We summarize these findings in the following theorem. Definitely well worth it, great app teaches me how to do math equations better than my teacher does and for that I'm greatful, I don't use the app to cheat I use it to check my answers and if I did something wrong I could get tough how to. do. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. What is the arc length of #f(x) = sinx # on #x in [pi/12,(5pi)/12] #? How do you find the arc length of the curve #y=e^(-x)+1/4e^x# from [0,1]? For \( i=0,1,2,,n\), let \( P={x_i}\) be a regular partition of \( [a,b]\). What I tried: a b ( x ) 2 + ( y ) 2 d t. r ( t) = ( t, 1 / t) 1 2 ( 1) 2 + ( 1 t 2) 2 d t. 1 2 1 + 1 t 4 d t. However, if my procedure to here is correct (I am not sure), then I wanted to solve this integral and that would give me my solution. 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 \(\PageIndex{2}\). What is the arc length of #f(x)=-xln(1/x)-xlnx# on #x in [3,5]#? #{dy}/{dx}={5x^4)/6-3/{10x^4}#, So, the integrand looks like: Use the process from the previous example. Do math equations . The distance between the two-p. point. OK, now for the harder stuff. in the x,y plane pr in the cartesian plane. 99 percent of the time its perfect, as someone who loves Maths, this app is really good! When \(x=1, u=5/4\), and when \(x=4, u=17/4.\) This gives us, \[\begin{align*} ^1_0(2\sqrt{x+\dfrac{1}{4}})dx &= ^{17/4}_{5/4}2\sqrt{u}du \\[4pt] &= 2\left[\dfrac{2}{3}u^{3/2}\right]^{17/4}_{5/4} \\[4pt] &=\dfrac{}{6}[17\sqrt{17}5\sqrt{5}]30.846 \end{align*}\]. How do you find the arc length of the curve #y = sqrt( 2 x^2 )#, #0 x 1#? a = rate of radial acceleration. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step What is the arclength of #f(x)=(1-x^(2/3))^(3/2) # in the interval #[0,1]#? If we now follow the same development we did earlier, we get a formula for arc length of a function \(x=g(y)\). Let \( f(x)=x^2\). Note that the slant height of this frustum is just the length of the line segment used to generate it. The same process can be applied to functions of \( y\). Round the answer to three decimal places. How do you find the arc length of the curve #y=sqrt(x-3)# over the interval [3,10]? http://mathinsight.org/length_curves_refresher, Keywords: What is the arc length of #f(x)= xsqrt(x^3-x+2) # on #x in [1,2] #? What is the arc length of #f(x) = x^2e^(3x) # on #x in [ 1,3] #? What is the arc length of #f(x)=2x-1# on #x in [0,3]#? Find the length of a polar curve over a given interval. What is the arc length of #f(x)=1/x-1/(5-x) # in the interval #[1,5]#? What is the arc length of the curve given by #r(t)=(4t,3t-6)# in the interval #t in [0,7]#? What is the arc length of #f(x)= e^(4x-1) # on #x in [2,4] #? \nonumber \], Adding up the lengths of all the line segments, we get, \[\text{Arc Length} \sum_{i=1}^n\sqrt{1+[f(x^_i)]^2}x.\nonumber \], This is a Riemann sum. Your IP: What is the arclength of #f(x)=(x-3)e^x-xln(x/2)# on #x in [2,3]#? To embed this widget in a post, install the Wolfram|Alpha Widget Shortcode Plugin and copy and paste the shortcode above into the HTML source. If you're looking for support from expert teachers, you've come to the right place. #sqrt{1+(frac{dx}{dy})^2}=sqrt{1+[(y-1)^{1/2}]^2}=sqrt{y}=y^{1/2}#, Finally, we have How do you find the length of the curve #x^(2/3)+y^(2/3)=1# for the first quadrant? S3 = (x3)2 + (y3)2 Here, we require \( f(x)\) to be differentiable, and furthermore we require its derivative, \( f(x),\) to be continuous. For objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces. \end{align*}\], Using a computer to approximate the value of this integral, we get, \[ ^3_1\sqrt{1+4x^2}\,dx 8.26815. \[y\sqrt{1+\left(\dfrac{x_i}{y}\right)^2}. Let \(g(y)=1/y\). altitude $dy$ is (by the Pythagorean theorem) What is the arclength of #f(x)=sqrt(4-x^2) # in the interval #[-2,2]#? How do you find the arc length of the curve # f(x)=e^x# from [0,20]? What is the arc length of #f(x)=x^2-2x+35# on #x in [1,7]#? arc length of the curve of the given interval. Looking for a quick and easy way to get detailed step-by-step answers? Let \(f(x)=(4/3)x^{3/2}\). How do you find the arc length of the curve #x=y+y^3# over the interval [1,4]? f ( x). \end{align*}\], Let \(u=x+1/4.\) Then, \(du=dx\). In this section, we use definite integrals to find the arc length of a curve. How do you find the lengths of the curve #y=(x-1)^(2/3)# for #1<=x<=9#? $$\hbox{ hypotenuse }=\sqrt{dx^2+dy^2}= What is the arclength of #f(x)=sqrt((x-1)(x+2)-3x# on #x in [1,3]#? How do you find the arc length of the curve #y = 2x - 3#, #-2 x 1#? We can write all those many lines in just one line using a Sum: But we are still doomed to a large number of calculations! How do you find the distance travelled from t=0 to #t=pi# by an object whose motion is #x=3cos2t, y=3sin2t#? Find the arc length of the function below? What is the arc length of #f(x) = ln(x) # on #x in [1,3] #? 5 stars amazing app. Perform the calculations to get the value of the length of the line segment. We study some techniques for integration in Introduction to Techniques of Integration. Let \( f(x)=\sqrt{1x}\) over the interval \( [0,1/2]\). This calculator calculates the deflection angle to any point on the curve(i) using length of spiral from tangent to any point (l), length of spiral (ls), radius of simple curve (r) values. The distance between the two-point is determined with respect to the reference point. You can find the. First, divide and multiply yi by xi: Now, as n approaches infinity (as wehead towards an infinite number of slices, and each slice gets smaller) we get: We now have an integral and we write dx to mean the x slices are approaching zero in width (likewise for dy): And dy/dx is the derivative of the function f(x), which can also be written f(x): And now suddenly we are in a much better place, we don't need to add up lots of slices, we can calculate an exact answer (if we can solve the differential and integral). 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 \(\PageIndex{2}\). A real world example. How do you find the arc length of the curve #f(x)=coshx# over the interval [0, 1]? What is the arc length of #f(x)= sqrt(5x+1) # on #x in [0,2]#? The cross-sections of the small cone and the large cone are similar triangles, so we see that, \[ \dfrac{r_2}{r_1}=\dfrac{sl}{s} \nonumber \], \[\begin{align*} \dfrac{r_2}{r_1} &=\dfrac{sl}{s} \\ r_2s &=r_1(sl) \\ r_2s &=r_1sr_1l \\ r_1l &=r_1sr_2s \\ r_1l &=(r_1r_2)s \\ \dfrac{r_1l}{r_1r_2} =s \end{align*}\], Then the lateral surface area (SA) of the frustum is, \[\begin{align*} S &= \text{(Lateral SA of large cone)} \text{(Lateral SA of small cone)} \\[4pt] &=r_1sr_2(sl) \\[4pt] &=r_1(\dfrac{r_1l}{r_1r_2})r_2(\dfrac{r_1l}{r_1r_2l}) \\[4pt] &=\dfrac{r^2_1l}{r^1r^2}\dfrac{r_1r_2l}{r_1r_2}+r_2l \\[4pt] &=\dfrac{r^2_1l}{r_1r_2}\dfrac{r_1r2_l}{r_1r_2}+\dfrac{r_2l(r_1r_2)}{r_1r_2} \\[4pt] &=\dfrac{r^2_1}{lr_1r_2}\dfrac{r_1r_2l}{r_1r_2} + \dfrac{r_1r_2l}{r_1r_2}\dfrac{r^2_2l}{r_1r_3} \\[4pt] &=\dfrac{(r^2_1r^2_2)l}{r_1r_2}=\dfrac{(r_1r+2)(r1+r2)l}{r_1r_2} \\[4pt] &= (r_1+r_2)l. \label{eq20} \end{align*} \]. How do you find the arc length of #x=2/3(y-1)^(3/2)# between #1<=y<=4#? What is the arc length of #f(x)=xe^(2x-3) # on #x in [3,4] #? Then, the surface area of the surface of revolution formed by revolving the graph of \(f(x)\) around the x-axis is given by, \[\text{Surface Area}=^b_a(2f(x)\sqrt{1+(f(x))^2})dx \nonumber \], Similarly, let \(g(y)\) be a nonnegative smooth function over the interval \([c,d]\). To find the surface area of the band, we need to find the lateral surface area, \(S\), of the frustum (the area of just the slanted outside surface of the frustum, not including the areas of the top or bottom faces). These findings are summarized in the following theorem. First, find the derivative x=17t^3+15t^2-13t+10, $$ x \left(t\right)=(17 t^{3} + 15 t^{2} 13 t + 10)=51 t^{2} + 30 t 13 $$, Then find the derivative of y=19t^3+2t^2-9t+11, $$ y \left(t\right)=(19 t^{3} + 2 t^{2} 9 t + 11)=57 t^{2} + 4 t 9 $$, At last, find the derivative of z=6t^3+7t^2-7t+10, $$ z \left(t\right)=(6 t^{3} + 7 t^{2} 7 t + 10)=18 t^{2} + 14 t 7 $$, $$ L = \int_{5}^{2} \sqrt{\left(51 t^{2} + 30 t 13\right)^2+\left(57 t^{2} + 4 t 9\right)^2+\left(18 t^{2} + 14 t 7\right)^2}dt $$. \end{align*}\], Using a computer to approximate the value of this integral, we get, \[ ^3_1\sqrt{1+4x^2}\,dx 8.26815. f (x) from. Click to reveal Wolfram|Alpha Widgets: "Parametric Arc Length" - Free Mathematics Widget Parametric Arc Length Added Oct 19, 2016 by Sravan75 in Mathematics Inputs the parametric equations of a curve, and outputs the length of the curve. How do you find the length of the curve defined by #f(x) = x^2# on the x-interval (0, 3)? $$ L = \int_a^b \sqrt{\left(x\left(t\right)\right)^2+ \left(y\left(t\right)\right)^2 + \left(z\left(t\right)\right)^2}dt $$. How do you find the arc length of the curve #y = 4 ln((x/4)^(2) - 1)# from [7,8]? 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 dy / dx is the derivative of the function y = f (x) with respect to x. This almost looks like a Riemann sum, except we have functions evaluated at two different points, \(x^_i\) and \(x^{**}_{i}\), over the interval \([x_{i1},x_i]\). Dont forget to change the limits of integration. We know the lateral surface area of a cone is given by, \[\text{Lateral Surface Area } =rs, \nonumber \]. The basic point here is a formula obtained by using the ideas of Let \( f(x)=\sin x\). 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