Question: Find the volume of the solid obtained
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
Transcribed Image Text:
y = x, y = Vx; about x = 2
> ABCD is a square piece of paper with sides of length 1 m. A quarter-circle is drawn from B to D with center A. The piece of paper is folded along EF, with E on AB and A on AD, so that A falls on the quarter-circle. Determine the maximum and minimum areas
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell. у 3х, у —2 — х?; about x — 1
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell. у%3D 4х — х, у — 3; about x %31
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell.
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell. y = x', y = 0, x = 1; about x = 2
> Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. x+у-3, х%4- (у — 1)2
> (a). Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 12. (i). L6 (sample points are left endpoints) (ii). R6 (sample points are right endpoints) (iii). M6 (sample points are midpoints) (b). Is L
> Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. x= 1+ (y – 2), x=2
> Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. x= Vỹ, x= 0, y=1
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. у%3D2 — х, у — 0, х — 1, х — 2;B about the x-axis
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x, y = Vr; about y = 1
> For what values of is there a straight line that intersects the curve in four distinct points? Y = x4 + cx3 + 12x2 – 5x + 2
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y =x', x = 2, y = 0; about the y-axis
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y? = x, x = 2y; about the y-axis
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = ¿r², y = 5 – x²; about the x-axis
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x', y = x, x> 0; about the x-axis
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = In x, y = 1, y = 2, x= 0; about the y-axis
> Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm
> Show that if a > -1 and b > a + 1, then the following integral is convergent. dx Jo 1+ x*
> Graph the astroid x = a cos3θ, y = a sin3θ and set up an integral for the area that it encloses. Then use a computer algebra system to evaluate the integral.
> Find the volume of the described solid S. A frustum of a pyramid with square base of side b, square top of side a, and height h. a b
> Find the volume of the described solid S. A cap of a sphere with radius r and heigh h. h
> For what values of does the curve y = cx3 + ex have inflection points?
> Investigate the family of functions f (x) = ln (sinx + C). What features do the members of this family have in common? How do they differ? For which values of C is f continuous on (-∞, ∞)? For which values of C does f have no graph at all? What happens a
> Find the volume of the described solid S. A frustum of a right circular cone with height h, lower base radius R, and top radius r. -r- h R
> Find the volume of the described solid S. A right circular cone with height h and base radius r.
> (a). A model for the shape of a bird’s egg is obtained by rotating about the -axis the region under the graph of Use a CAS to find the volume of such an egg. (b). For a Red-throated Loon, a = -0.06, b = 0.04, c = 0.1, and d = 0.54. Gr
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. x = 2/y, x= 0, y = 9; about the y-axis %3D
> (a). If the region shown in the figure is rotated about the x-axis to form a solid, use Simpson’s Rule with n = 8 to estimate the volume of the solid. (b). Estimate the volume if the region is rotated about the y-axis. Use Simpson&aci
> A log 10 m long is cut at 1-meter intervals and its cross-sectional areas A (at a distance from the end of the log) are listed in the table. Use the Midpoint Rule with n = 5 to estimate the volume of the log. x (m) A (m²) x (m) A (m²) 0.68 6. 0.53 1
> A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm l
> Suppose f is continuous on [0, ∞] and limx→∞f (x) = 1. Is it possible that f∞0f (x) dx is convergent?
> Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building 120 ft high. (a). How much work is done in pulling the rope
> Each integral represents the volume of a solid. Describe the solid. (a) 7 y dy (b) 7 [(1 (1 + cos x)? – 1°]dx
> For any number c, we let fc (x) be the smaller of the two numbers (x – c)2 and (x – c – 2)2. Then we define Find the maximum and minimum values of g (c) if -2 g(c) f.(x)dx
> Each integral represents the volume of a solid. Describe the solid. *w/2 (а) т cos?x dx (b) (y* – y*) dy
> Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x, y = xel-/2, about y = 3
> Use a graph to find approximate -coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the -axis the region bounded by these curves. y = 3 s
> Use a graph to find approximate -coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the -axis the region bounded by these curves. y = 2 +
> Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y = cos x, y = 2 – cos x, 0<I< 27; about y 4
> Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. x² – y? = 1, x= 3; about x = -2
> The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. y = x', y = Vx; about y = 1
> The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. y = x', y = Vx; about x = 1
> The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. x= y?, x = 1; about x = 1
> Use Newton’s method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. 3 sin(x?) = 2x
> When gas expands in a cylinder with radius r, the pressure at any given time is a function of the volume: P = P (V). The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: F = πr2p. Show that t
> The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. х — у— 1, у— х? — 4х + 3; about y— 3
> The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. x = 2y – y', x = 0; about the y-axis
> The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. у 3 1/х, х 3D 1, х— 2, у %3D 0; about the x-axis
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. у 3D1+ sec x, yу 3 3; about y —1
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = e", y = 1, x = 2; about y = 2
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. x=1- y, x= y² – 1
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. у %3 х? — 2х, у%3х+4
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = x², y? = x
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. у 3 In x, ху — 4, х%3D1, х— 3
> A spinner from a board game randomly indicates a real number between 0 and 10. The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length.
> Let g (x) = fx0f (t) dt, where f is the function whose graph is shown. (a). Evaluate g (0) and g (6). (b). Estimate g (x) for x = 1, 2, 3, 4, and 5. (c). On what interval g is increasing? (d). Where does have a maximum value? (e). Sketch a rough graph of
> Find the values of p for which the integral converges and evaluate the integral for those values of p. dx
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = e", y = x? – 1, x=-1, x = 1 x = -1,
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. sin?x - dx V- x,
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. sec'x -dx lo xVx
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. 2 + e -
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. x' + 1 .3
> Show that 1/3 Tn + 3.2Mn = S2n.
> Find the number b such that the line y = b divides the region bounded by the curves y = x2 and y = 4 into two regions with equal area.
> Sketch the region and find its area (if the area is finite). s = {(x, y) | –2 < r< 0, 0 < y< 1//I + 2 }
> A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surfa
> Find the area of the shaded region. YA x= y? - 4y (-3, 3) x= 2y - y?
> Let g (x) = fx0 f (t) dt, where f is the function whose graph is shown. (a). Evaluate g (x) for x = 0, 1, 2, 3, 4, 5 and 6. (b). Estimate g (7). (c). Where does b have a maximum value? Where does it have a minimum value? (d). Sketch a rough graph of g.
> Sketch the region and find its area (if the area is finite). S = {(x, y) | 0 < x < n/2, 0 < y< sec?r}
> Sketch the region and find its area (if the area is finite). S = {(x, y) | x > 0, 0 < y< xe*}
> Sketch the region and find its area (if the area is finite). S = {(x, y) | 0 < y < 2/(x² + 9)}
> Find the area enclosed by the x-axis and the curve x = 1 + et, y = t – t2.
> Find the area enclosed by the curve x = t2 – 2t, y = √t and the y-axis.
> Use the parametric equations of an ellipse, x = a cos θ, y = b sin θ, 0 < θ < 2π, to find the area that it encloses.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. csc x dx J2/2
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. dx e - 1
> Graph f (x) = sin (ex) and use the graph to estimate the value of t such that ft+1tf (x) dx is a maximum. Then find the exact value of t that maximizes this integral.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 dy 4y – 1 Je
> Find the area of the shaded region. yA x= y?- 2 y=1 x= e y=-
> A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is a (t) = 60t, at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (d
> If the birth rate of a population is b (t) = 2200e0.024t people per year and the death rate is d (t) = 1460e0.018t people per year, find the area between these curves for 0 < t < 10. What does this area represent?
> A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are 5.8, 20.3, 26.7, 29.0, 29.0, 27.6, 26.3, 28.8, 20.5, 15.1, 8.7 and 2.8. Use Simpson’s Rule to estima
> The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use Simpson’s Rule to estimate the area of the pool. 5.6 5.0 6.8 4.8 4.8 7.2 6.2
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 dx 3— х
> Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use Simpson’s Rule to estimate how much farther Kell
> Sketch the curves y = cos x and y = 1 – cos x, 0 < x < π, and observe that the region between them consists of two separate parts. Find the area of this region.
> Sketch the region that lies between the curves y = cos x and y = sin 2x and between x = 0 and x = π/2. Notice that the region consists of two separate parts. Find the area of this region.
> Show that f10(1 – x2) n dx = 22n(n!)2/ (2n + 1)! Hint: Start by showing that if denotes the integral, then 2k + 2 - Ik 2k + 3 Ik+1
> Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = x cos x, y=x10
> Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = x² In x, y = VI – I (х — 1
> Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = y = x* – x, x 0 (x² + 1)*"
> A high-speed bullet train accelerates and decelerates at the rate of 4ft/s2. Its maximum cruising speed is 90 mi/h. (a). What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at th
> Find the area of the shaded region. y. y = Vx+2 x= 2 1 х+1
> Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = x sin(x²), y = x*
> Sketch the region enclosed by the given curves and find its area. y = 3x², y= 8r², 4x + y = 4, x> 0 %3D
> Sketch the region enclosed by the given curves and find its area. y = 1/x, y= x, y=}x, x>0
> Sketch the region enclosed by the given curves and find its area. у %3D сos x, у — 2 — cos x, 0x<2п
> Sketch the region enclosed by the given curves and find its area. y = e", y = xe", x= 0
> A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until chambers, including the sphere, are formed. (The figure shows the case n = 4.) Use mathema
