Question: Formula 4 is valid only when f(
Formula 4 is valid only when f(x) > 0. Show that when f(x) is not necessarily positive, the formula for surface area becomes
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s = [" 27| f(x)|/T +[S(x)F dx
> (a) Graph the functions f(x) = 1/x1.1 and t(x) = 1/x0.9 in the viewing rectangles [0, 10] by [0, 1] and [0, 100] by [0, 1]. (b) Find the areas under the graphs of f and t from x = 1 to x = t and evaluate for t = 10, 100, 104, 106, 1010, and 1020. (c) Fin
> Find the area under the curve y = 1/x3 from x = 1 to x = t and evaluate it for t = 10, 100, and 1000. Then find the total area under this curve for x > 1.
> Which of the following integrals are improper? Why? (a) " tan x dx (b) " tan x dx dx (c) ix? (d) fe*dx 2 – x – 2
> Explain why each of the following integrals is improper. (a) f -dx 1 1 - dx Jo 1 + x³ (b) X - (c) x'e * dx cot x dx
> The curves with equations xn + yn = 1, n = 4, 6, 8, . . . , are called fat circles. Graph the curves with n = 2, 4, 6, 8, and 10 to see why. Set up an integral for the length L2k of the fat circle with n = 2k. Without attempting to evaluate this integra
> Find the length of the curve yーfp- 13 – 1 dt 1<x< 4
> (a) The figure shows a telephone wire hanging between two poles at x 2b and x b. It takes the shape of a catenary with equation y c + a cosh (x/a). Find the length of the wire. (b) Suppose two telephone poles are 50 ft apart and the length of
> Find a power series representation for the function and determine the interval of convergence. f(x) 2x + 3 4-
> The Gateway Arch in St. Louis (see the photo on page 543) was constructed using the equation for the central curve of the arch, where x and y are measured in meters and |x| y = 211.49 – 20.96 cosh 0.03291765x
> A hawk flying at 15 m/s at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation until it hits the ground, where y is its height above the ground and x is the horizontal distance trave
> A steady wind blows a kite due west. The kite’s height above ground from horizontal position x = 0 to x = 80 ft is given by / Find the distance traveled by the kite.
> The arc length function for a curve y = f x), where f is an increasing function, is / (a) If f has y intercept 2, find an equation for f. (b) What point on the graph off is 3 units along the curve from the y intercept? State your answer rounded to 3 de
> Find the arc length function for the curve y = sin 'x + V1 – x² with starting point (0, 1)
> (a) Find the arc length function for the curve y = In(sin x), (b) Graph both the curve and its arc length function on the same screen. 0<x< T, with starting point (7/2, 0)
> Find the arc length function for the curve y = 2x3/2 with starting point P0(1, 2).
> (a) Sketch the curve y3 = x2. (b) Use Formulas 3 and 4 to set up two integrals for the arc length from (0, 0) to (1, 1). Observe that one of these is an improper integral and evaluate both of them. (c) Find the length of the arc of this curve from (21, 1
> Sketch the curve with equation x2/3 + y2/3 = 1 and use symmetry to find its length.
> Find a power series representation for the function and determine the interval of convergence. 2 S(x) 3 — х
> Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve y = x4/3 that lies between the points (0, 0) and (1, 1). If your CAS has trouble evaluating the integral, make a substitution that changes the i
> Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve y = ex that lies between the points (0, 1) and (2, e2).
> Repeat Exercise 29 for the curve Data from Exercise 29: (a) Graph the curve / (b) Compute the lengths of inscribed polygons with n = 1, 2, and 4 sides. (Divide the interval into equal sub intervals.) Illustrate by sketching these polygons. (c) Set up a
> Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator. y = e *, 0<x< 2
> Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator. y = In(1 + x³), 0 <x< 5
> Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator. y = Vx, 1<x< 6
> Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator. y = x sin x, 0 <x< 27
> Graph the curve and visually estimate its length. Then use your calculator to find the length correct to four decimal places. y = x + cos x, 0<x<T/2
> Graph the curve and visually estimate its length. Then use your calculator to find the length correct to four decimal places. y = x² + x', 1 <x<2
> Find a power series representation for the function and determine the interval of convergence. 5 S(x) 1 – 4x?
> (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor’s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (c) Check your result in part (b) by graphing / S
> Find the length of the arc of the curve from point P to point Q. х? — (у — 4), Р(1, 5), Q(8, 8)
> Find the length of the arc of the curve from point P to point Q. y = }x', P(-1, ), e(1,4)
> Find the exact length of the curve. y = 1- e, 0<x<2
> Find the exact length of the curve. y = In(1 – x'), 0 <x<}
> Find the exact length of the curve. у — Vx (x) y = — х2 + sin
> Find the exact length of the curve. y = x? - } In x, 1<x<2
> Find the exact length of the curve. y = 3 + cosh 2.x, 0<x<1
> Find the exact length of the curve. у — In(sec x), 0<x< п/4 y
> Find the exact length of the curve. y = In(cos x), 0 <x</3 %3|
> Find the exact length of the curve. 3 Vy (у — 3), 1<y<9
> Find a power series representation for the function and determine the interval of convergence. 1 1+ x S(x)
> Find the exact length of the curve. 1 1<y<2 8 4y² +
> Find the exact length of the curve. 1<x<2 y = 3 4x +
> Find the exact length of the curve. 36y? = (x² – 4), 2<x<3, y > 0 %3D
> Find the exact length of the curve. y = 1 + 6x2, 0<x<1 3/2
> Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. y? = In x, -1 < y< 1
> Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. x= Vy - y, 1< y< 4 X =
> Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. y = xe, 0<I<2
> Use the arc length formula to find the length of the curve / Check your answer by noting that the curve is part of a circle.
> Use the arc length formula (3) to find the length of the curve y = 2x - 5, 21 < x < 3. Check your answer by noting that the curve is a line segment and calculating its length by the distance formula.
> Let L be the length of the curve y = f(x), a < x < b, where f is positive and has a continuous derivative. Let Sf be the surface area generated by rotating the curve about the x axis. If c is a positive constant, define t(x) = f(x) 1 c and let St be the
> Show that if we rotate the curve y = ex/2 + ex/2 about the x axis, the area of the resulting surface is the same value as the enclosed volume for any interval a < x < b.
> Find the area of the surface obtained by rotating the circle x2 + y2 = r2 about the line y = r.
> Use the result of Exercise 33 to set up an integral to find the area of the surface generated by rotating the curve / 0 < x < 4, about the line y = 4. Then use a CAS to evaluate the integral. Data from Exercise 33: If the curve y = f(x), a < x < b, is
> If the curve y = f(x), a < x < b, is rotated about the horizontal line y = c, where f(x) < c, find a formula for the area of the resulting surface.
> Find the surface area of the torus in Exercise 6.2.63. Data from Exercise 6.2.63: (a) Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii r and R. (b) By interpreting the integral as an area, find
> A group of engineers is building a parabolic satellite dish whose shape will be formed by rotating the curve y = ax2 about the y axis. If the dish is to have a 10ft diameter and a maximum depth of 2 ft, find the value of a and the surface area of the d
> (a) If a > 0, find the area of the surface generated by rotating the loop of the curve 3ay2 = x (a - x)2 about the x axis. (b) Find the surface area if the loop is rotated about the y axis.
> If the infinite curve y = ex, x > 0, is rotated about the x axis, find the area of the resulting surface.
> Use a CAS to find the exact area of the surface obtained by rotating the curve about the y axis. If your CAS has trouble evaluating the integral, express the surface area as an integral in the other variable. у — In(x + 1), 0 <x<1 y
> Use a CAS to find the exact area of the surface obtained by rotating the curve about the y­ axis. If your CAS has trouble evaluating the integral, express the surface area as an integral in the other variable. y = x', 0< y< 1
> Use either a CAS or a table of integrals to find the exact area of the surface obtained by rotating the given curve about the x­ axis y = Vx? + 1, 0<x<3
> Use either a CAS or a table of integrals to find the exact area of the surface obtained by rotating the given curve about the x­ axis. y = 1/x, 1<x< 2
> Use Simpson’s Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x axis. Compare your answer with the value of the integral produced by a calculator. y = x In x, 1 r< 2
> Use Simpson’s Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x axis. Compare your answer with the value of the integral produced by a calculator. y = xe", 0 < x<1
> Use Simpson’s Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x axis. Compare your answer with the value of the integral produced by a calculator. y = x + x², 0<x<1
> Use Simpson’s Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x­ axis. Compare your answer with the value of the integral produced by a calculator. y = x', 0< x < 5
> The given curve is rotated about the y axis. Find the area of the resulting surface. y = x' - } In x, 1 < x< 2
> The given curve is rotated about the y axis. Find the area of the resulting surface. x = Va? – y², 0 <y<a/2
> The given curve is rotated about the y axis. Find the area of the resulting surface. x2/3 + y/3 = 1, 0< y<1 %3|
> The given curve is rotated about the y­ axis. Find the area of the resulting surface. y = }x2, 0 <x< 12
> Find the exact area of the surface obtained by rotating the curve about the x axis. x = 1 + 2y?, 1 <y<2
> Find the exact area of the surface obtained by rotating the curve about the x axis. x = }(y² + 2)/², 1<y<2 + 2)/2, ミys2
> Find the exact area of the surface obtained by rotating the curve about the x axis. y = 6. 2x
> Find the exact area of the surface obtained by rotating the curve about the x axis. y = cos(}x), 0<x<T
> Find the exact area of the surface obtained by rotating the curve about the x axis. y = V1 + e*, 0<x<1
> Find the exact area of the surface obtained by rotating the curve about the x axis. y² = x + 1, 0<x< 3
> Find the exact area of the surface obtained by rotating the curve about the x axis. у3 V5 — х, 3<x<5
> Find the exact area of the surface obtained by rotating the curve about the x­ axis. y = x', 0< x < 2
> (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x axis and (ii) the y axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. у —
> (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x axis and (ii) the y axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. x =
> (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x axis and (ii) the y axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. x =
> (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x axis and (ii) the y axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. y =
> (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x axis and (ii) the y axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. y =
> (a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x­ axis and (ii) the y­ axis. (b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four
> Evaluate the integral using integration by parts with the indicated choices of u and dv. хе ^ dx; и —х, dv — — х, do — e 2* dx
> Let 5 be the region that lies between the curves where m and n are integers with 0 (a) Sketch the region /. (b) Find the coordinates of the centroid of /. (c) Try to find values of m and n such that the centroid lies outside /. y = x" y = x" 0 < x< 1
> Use the Second Theorem of Pappus described in Exercise 48 to find the surface area of the torus in Example 7.
> The Second Theorem of Pappus is in the same spirit as Pappus’s Theorem on page 565, but for surface area rather than volume: Let C be a curve that lies entirely on one side of a line l in the plane. If C is rotated about l, then the area of the resulting
> The centroid of a curve can be found by a process similar to the one we used for finding the centroid of a region. If C is a curve with length L, then the centroid is (x, y) where / Here we assign appropriate limits of integration, and ds is as defined i
> Use the Theorem of Pappus to find the volume of the given solid. The solid obtained by rotating the triangle with vertices (2, 3), (2, 5), and (5, 4) about the x-axis
> Use the Theorem of Pappus to find the volume of the given solid. A cone with height h and base radius r
> Use the Theorem of Pappus to find the volume of the given solid. A sphere of radius r
> If x is the x-coordinate of the centroid of the region that lies under the graph of a continuous function f, where a (ex + d)f(x) dx = (cã + d) [* f(x) dx %3D
> A rectangle R with sides a and b is divided into two parts R1 and R2 by an arc of a parabola that has its vertex at one corner of R and passes through the opposite corner. Find the centroids of both R1and R2. R2 Ry
> Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles and using additivity of moments. -2 i 2 X -1 2.
> Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles and using additivity of moments. yA 2 1- -1 1 2.
> Prove that the centroid of any triangle is located at the point of intersection of the medians.
> Use a graph to find approximate x-coordinates of the points of intersection of the curves y = ex and y = 2 - x2. Then find (approximately) the centroid of the region bounded by these curves.
> Find the centroid of the region bounded by the curves y = x3 - x and y = x2 - 1. Sketch the region and plot the centroid to see if your answer is reasonable.
> Use Simpson’s Rule to estimate the centroid of the region shown. y 4 4 8 x 2. 2.
