Question: When gas expands in a cylinder with
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 the work done by the gas when the volume expands from volume V1 to volume V2 is
Transcribed Image Text:
w = * P dv piston head
> 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 = 1- x², y = 0; about the x-axis
> 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 axis. x = /sin y, 0 sy<T, I= 0; about y = 4
> 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', x = y?; about y = -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. y = x', y = 0, x= 1; about y = 1
> 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
> 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. y = x, y = Vx; about x = 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. 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
