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 that the triangle AEF can have.
> Find the average value of the function on the given interval. fa) — 4х — х, [О, 4]
> Find the exact length of the curve. x = y3/2, 0 < y < 1
> Find the exact length of the curve. y2 = 4 (x + 4)3, 0 < x < 2, y > 0
> Find the exact length of the curve. x = 1 + 3t2, y = 4 + 2t3, 0 < t < 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 = t cos t, y =t sin t, 0 st< 27 0 <t< 27
> The figure shows a region consisting of all points inside a square that are closer to the center than to the sides of the square. Find the area of the region. 2. 2. 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. x =t + cos t, y=t- sin t, 0st< 2m
> Let f (x) = 0 if is any rational number and f (x) = 1 if is any irrational number. Show that f is not integrable on [0, 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= y? – 2y, 0 <y<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 = sin x, 0<x<T
> If the region shown in the figure is rotated about the y-axis to form a solid, use Simpson’s Rule with n = 10 to estimate the volume of the solid. y 5 4 1 1 2 3 4 5 6 7 8 9 10 11 12 í 3. 2.
> Use Simpson’s Rule with n = 10 to estimate the volume obtained by rotating about the y-axis the region under the curve y = √1 + x3, 0 < x < 1.
> Find the length of the loop of the curve x = 3t – t3, y = 3t2.
> (a). In Example 2 in Section 1.7 we showed that the parametric equations x = cos t, y = sin t, 0 < t < 2π, represent the unit circle. Use these equations to show that the length of the unit circle has the expected value. (b). In Example 3 in Section 1.7
> 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 your calculator. x = sin t, y = 1?, 0 st< 2n
> 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 your calculator. y = xe, 0<I<5
> Sketch the region in the plane consisting of all points (x, y) such that 2xy < |x – y|< x? + y?
> Graph the curve and find its exact length. x = et + e-t, y = 5 – 2t, 0 < t < 3
> Graph the curve and find its exact length. x = et cos t, y = et sin t, 0 < t < π
> On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the je
> Graph the curve and find its exact length. y = x3/3 + 1/4x, 1 < x < 2
> Graph the curve and find its exact length. x = et – t, y = 4et/2, -8 < t < 3
> Find the exact length of the curve. x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ), 0 < θ < π
> Find the exact length of the curve. y = 1/4x2 – 1/2 lnx, 1 < x < 2
> Find the exact length of the curve. y = √x – x2 + sin-1 (√x)
> Use the arc length formula (2) to find the length of the curve y – 2x - 5, -1 < x < 3. Check your answer by noting that the curve is a line segment and calculating its length by the distance formula.
> 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. х%3D1+y;, х30, у%31, у%3D2
> A cone of radius r centimeters and height h centimeters is lowered point first at a rate of 1 cm/s into a tall cylinder of radius R centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely
> Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by y = √x and y = x2. Find V both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. Sketch the region and a typical shell. y = 4(x – 2), y =x² – 4x + 7
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. Sketch the region and a typical shell. y = 3 + 2x – x², x+ y = 3
> Economists use a cumulative distribution called a Lorenz curve to describe the distribution of income between households in a given country. Typically, a Lorenz curve is defined on [0, 1] with endpoints (0, 0) and (1, 1), and is continuous, increasing, a
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. Sketch the region and a typical shell. y = e*, y = 0, x= 0, x= 1
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. Sketch the region and a typical shell. y = x², y = 0, x= 1
> The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. x² + (y – 1)? = 1; about the y-axis
> The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. y = -x? + 6x – 8, y = 0; about the x-axis
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. Sketch the region and a typical shell. у%3D 1/х, у—0, х— 1, х—2
> 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'sin x, y = 0, 0 <x< T; about x = -1
> Given a sphere with radius r, find the height of a pyramid of minimum volume whose base is a square and whose base and triangular face are all tangent to the sphere. What if the base of the pyramid is a regular n-gon? (A regular n-gon is a polygon with e
> 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 = sin?x, y = sin*x, 0 <x< T; about x = T/2
> Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the -axis the region enclosed by these curves.
> Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the -axis the region enclosed by these curves.
> Each integral represents the volume of a solid. Describe the solid. y dy 1+ y* (a) 27 (b) * 27 (7 – x)(cos x – sin x) dx
> Explain exactly what is meant by the statement that “differentiation and integration are inverse processes.”
> 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. y = e*, y = 0, x = 0, x = 4; about x= 5
> 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
> 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. 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
