Solved: 8990. Average value Use the definition for the

Describe and sketch a region that is bounded above and below by two curves.

Describe and a sketch a region that is bounded on the left and on the right by two curves.

Which order of integration is preferable to integrate f 1x, y2 = xy over R = 51x, y2: y - 1 x 1 - y, 0 y 16?

Which order of integration would you use to find the area of the region bounded by the x-axis and the lines y = 2x + 3 and y = 3x - 4 using a double integral?

Change the order of integration in the integral 1 1 0 1 1y y2 f 1x, y2 dx dy.

Sketch the region of integration for 1 2 -21 4 x2 exy dy dx.

78. Regions of integration Consider the regions R shown in the figures and write an iterated integral of a continuous function f over R.

78. Regions of integration Consider the regions R shown in the figures and write an iterated integral of a continuous function f over R.

916. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dy dx. R = 51x, y2: 0 x p>4, sin x y cos x6

916. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dy dx. R = 51x, y2: 0 x 2, 3x2 y -6x + 246

916. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dy dx. R = 51x, y2: 1 x 2, x + 1 y 2x + 46

916. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dy dx. R = 51x, y2: 0 x 4, x2 y 81x6

916. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dy dx. R is the triangular region with vertices (0, 0), (0, 2), and (1, 0).

916. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dy dx. R is the triangular region with vertices (0, 0), (0, 2), and (1, 1).

916. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dy dx. R is the region in the first quadrant bounded by a circle of radius 1 centered at the origin.

916. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dy dx. R is the region in the first quadrant bounded by the y-axis and the parabolas y = x2 and y = 1 - x2.

1726. Evaluating integrals Evaluate the following integrals as they are written. L 1 0 L 1 x 6y dy dx

1726. Evaluating integrals Evaluate the following integrals as they are written. L 1 0 L 2x 0 15 xy2 dy dx

1726. Evaluating integrals Evaluate the following integrals as they are written. L 2 0 L 2x x2 xy dy dx

1726. Evaluating integrals Evaluate the following integrals as they are written. L 3 0 L x + 6 x2 1x - 12 dy dx

1726. Evaluating integrals Evaluate the following integrals as they are written. L p>4 -p>4 L cos x sin x dy dx

1726. Evaluating integrals Evaluate the following integrals as they are written. L 1 0 L 21 - x2 -21 - x2 2x2y dy dx

1726. Evaluating integrals Evaluate the following integrals as they are written. L 2 -2L 8-x2 x2 x dy dx

1726. Evaluating integrals Evaluate the following integrals as they are written. L ln 2 0 L 2 ex dy dx

1726. Evaluating integrals Evaluate the following integrals as they are written. L 1 0 L x 0 2ex2dy dx

1726. Evaluating integrals Evaluate the following integrals as they are written. L 23 p>2 0 L x 0 y cos x3 dy dx

2730. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 4R xy dA; R is bounded by x = 0, y = 2x + 1, and y = -2x + 5.

2730. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 4R 1x + y2 dA; R is the region in the first quadrant bounded by x = 0, y = x2, and y = 8 - x2.

2730. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 4R y2 dA; R is bounded by x = 1, y = 2x + 2, and y = -x - 1.

2730. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 4R x2y dA; R is the region in quadrants 1 and 4 bounded by the semicircle of radius 4 centered at (0, 0).

3132. Regions of integration Write an iterated integral of a continuous function f over the region R shown in the figure.

3132. Regions of integration Write an iterated integral of a continuous function f over the region R shown in the figure.

3338. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dx dy. The region bounded by y = 2x + 3, y = 3x - 7, and y = 0

3338. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dx dy. R = 51x, y2: 0 x y11 - y26

3338. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dx dy. The region bounded by y = 4 - x, y = 1, and x = 0

3338. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dx dy. The region in quadrants 2 and 3 bounded by the semicircle with radius 3 centered at (0, 0)

3338. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dx dy. The region bounded by the triangle with vertices (0, 0), (2, 0), and (1, 1)

3338. Regions of integration Sketch each region and write an iterated integral of a continuous function f over the region. Use the order dx dy. The region in the first quadrant bounded by the x-axis, the line x = 6 - y, and the curve y = 1x

3946. Evaluating integrals Evaluate the following integrals as they are written. L 2 -1L 4 - y y dx dy

3946. Evaluating integrals Evaluate the following integrals as they are written. L 2 0 L 4 - y2 0 y dx dy

3946. Evaluating integrals Evaluate the following integrals as they are written. L 4 0 L 216 - y2 -216 - y2 2xy dx dy

3946. Evaluating integrals Evaluate the following integrals as they are written. L 1 0 L 221 - y2 -221 - y2 2x dx dy

3946. Evaluating integrals Evaluate the following integrals as they are written. L ln 2 0 L 2 ey y x dx dy

3946. Evaluating integrals Evaluate the following integrals as they are written. L 4 0 L 2y y xy dx dy

3946. Evaluating integrals Evaluate the following integrals as they are written. L p>2 0 L p>2 y 6 sin 12x - 3y2 dx dy

3946. Evaluating integrals Evaluate the following integrals as they are written. L p>2 0 L cos y 0 esin y dx dy

4752. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 4R 12y dA; R is bounded by y = 2 - x, y = 1x, and y = 0.

4752. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 4R y2 dA; R is bounded by y = 1, y = 1 - x, and y = x - 1.

4752. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 4R 3xy dA; R is bounded by y = 2 - x, y = 0, and x = 4 - y2 in the first quadrant.

4752. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 4R 1x + y2 dA; R is bounded by y = _ x _ and y = 4.

4752. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 4R 3x2 dA; R is bounded by y = 0, y = 2x + 4, and y = x3.

4752. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 4R x2 y dA; R is bounded by y = 0, y = 1x, and y = x - 2.

5356. Volumes Use double integrals to calculate the volume of the following regions. The tetrahedron bounded by the coordinate planes 1x = 0, y = 0, z = 02 and the plane z = 8 - 2x - 4y

5356. Volumes Use double integrals to calculate the volume of the following regions. The solid in the first octant bounded by the coordinate planes and the surface z = 1 - y - x2

5356. Volumes Use double integrals to calculate the volume of the following regions. The segment of the cylinder x2 + y2 = 1 bounded above by the plane z = 12 + x + y and below by z = 0

5356. Volumes Use double integrals to calculate the volume of the following regions. The solid beneath the cylinder z = y2 and above the region R = 51x, y2: 0 y 1, y x 16

5762. Changing order of integration Reverse the order of integration in the following integrals. L 2 0 L 2x x2 f 1x, y2 dy dx y 1 x 1 0 R y _ 2x (2, 4) y _ x2

5762. Changing order of integration Reverse the order of integration in the following integrals. L 3 0 L 6-2x 0 f 1x, y2 dy dx y 1 x 1 0 R y _ 6 _ 2x

5762. Changing order of integration Reverse the order of integration in the following integrals. L 1 1>2L -ln y 0 f 1x, y2 dx dy

5762. Changing order of integration Reverse the order of integration in the following integrals. L 1 0 L ey 1 f 1x, y2 dx dy

5762. Changing order of integration Reverse the order of integration in the following integrals. L 1 0 L cos-1 y 0 f 1x, y2 dx dy

5762. Changing order of integration Reverse the order of integration in the following integrals. L e 1 L ln x 0 f 1x, y2 dy dx

6368. Changing order of integration The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. L 1 0 L 1 y ex2 dx dy

6368. Changing order of integration The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. L p 0 L p x sin y2 dy dx

6368. Changing order of integration The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. L 1>2 0 L 1>4 y2 y cos 116px22 dx dy

6368. Changing order of integration The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. L 4 0 L 2 1x x y5 + 1 dy dx

6368. Changing order of integration The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. L 23 p 0 L 23 p y x4 cos 1x2y2 dx dy

6368. Changing order of integration The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. L 2 0 L 4-x2 0 xe2y 4 - y dy dx

6974. Regions between surfaces Find the volume of the following solid regions. The solid above the region R = 51x, y2: 0 x 1, 0 y 1 - x6 bounded by the paraboloids z = x2 + y2 and z = 2 - x2 - y2, and the coordinate planes in the first octant y z x z _ x2 _ y2 z _ 2 _ x2 _ y2 R

6974. Regions between surfaces Find the volume of the following solid regions. The solid above the parabolic region R = 51x, y2: 0 x 1, 0 y 1 - x26 and between the planes z = 1 and z = 2 - y z _ 1 z x y R z _ 2 _ y

6974. Regions between surfaces Find the volume of the following solid regions. The solid bounded by the paraboloid z = x2 + y2 and the plane z = 9 x y z z _ 9 z _ x2 _ y2

6974. Regions between surfaces Find the volume of the following solid regions. The solid bounded by the parabolic cylinder z = x2 + 1, and the planes z = y + 1 and y = 1 z _x2 _ 1 z _y _ 1 y x z R

6974. Regions between surfaces Find the volume of the following solid regions. The solid above the region R = 51x, y2: 0 x 1, 0 y 2 - x6 and between the planes -4x - 4y + z = 0 and -2x - y + z = 8

6974. Regions between surfaces Find the volume of the following solid regions. The solid S between the surfaces z = ex - y and z = -ex - y, where S intersects the xy-plane in the region R = 51x, y2: 0 x y, 0 y 16 y x z z _ _ex _ y z _ ex _ y R

7580. Area of plane regions Use double integrals to compute the area of the following regions. Make a sketch of the region. The region bounded by the parabola y = x2 and the line y = 4

7580. Area of plane regions Use double integrals to compute the area of the following regions. Make a sketch of the region. The region bounded by the parabola y = x2 and the line y = x + 2

7580. Area of plane regions Use double integrals to compute the area of the following regions. Make a sketch of the region. The region in the first quadrant bounded by y = ex and x = ln 2

7580. Area of plane regions Use double integrals to compute the area of the following regions. Make a sketch of the region. The region bounded by y = 1 + sin x and y = 1 - sin x on the interval 30, p4

7580. Area of plane regions Use double integrals to compute the area of the following regions. Make a sketch of the region. The region in the first quadrant bounded by y = x2, y = 5x + 6, and y = 6 - x

7580. Area of plane regions Use double integrals to compute the area of the following regions. Make a sketch of the region. The region bounded by the lines x = 0, x = 4, y = x, and y = 2x + 1

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. In the iterated integral 1 d c 1 b a f 1x, y2 dx dy, the limits a and b must be constants or functions of x. b. In the iterated integral 1 d c 1 b a f 1x, y2 dx dy, the limits c and d must be functions of y. c. Changing the order of integration gives 1 2 0 1 y 1 f 1x, y2 dx dy = 1 y 1 1 2 0 f 1x, y2 dy dx.

8285. Miscellaneous integrals Evaluate the following integrals. OR y dA; R = 51x, y2: 0 y sec x, 0 x p>36

8285. Miscellaneous integrals Evaluate the following integrals. OR 1x + y2 dA; R is the region bounded by y = 1>x and y = 5>2 - x.

8285. Miscellaneous integrals Evaluate the following integrals. OR xy 1 + x2 + y2 dA; R = 51x, y2: 0 y x, 0 x 26

8285. Miscellaneous integrals Evaluate the following integrals. OR x sec2 y dA; R = 51x, y2: 0 y x2, 0 x 1p>26

Paraboloid sliced by plane Find the volume of the solid between the paraboloid z = x2 + y2 and the plane z = 1 - 2y.

Two integrals to one Draw the regions of integration and write the following integrals as a single iterated integral: L 1 0 L e ey f 1x, y2 dx dy + L 0 -1 L e e-y f 1x, y2 dx dy.

Square region Consider the region R = 51x, y2: _ x _ + _ y _ 16 shown in the figure. a. Use a double integral to verify that the area of R is 2. b. Find the volume of the square column whose base is R and whose upper surface is z = 12 - 3x - 4y. c. Find the volume of the solid above R and beneath the cylinder x2 + z2 = 1. d. Find the volume of the pyramid whose base is R and whose vertex is on the z-axis at (0, 0, 6). _1 1 _1 1 y x R _x_ _ _y_ _ 1

8990. Average value Use the definition for the average value of a function over a region R (Section 13.1), f = 1 area of R OR f 1x, y2 dA. Find the average value of a - x - y over the region R = 51x, y2: x + y a, x 0, y 06, where a 7 0.

8990. Average value Use the definition for the average value of a function over a region R (Section 13.1), f = 1 area of R OR f 1x, y2 dA. Find the average value of z = a2 - x2 - y2 over the region R = 51x, y2: x2 + y2 a26, where a 7 0.

9192. Area integrals Consider the following regions R. a. Sketch the region R. b. Evaluate 4R dA to determine the area of the region. c. Evaluate 4R xy dA. R is the region between both branches of y = 1>x and the lines y = x + 3>2 and y = x - 3>2.

9192. Area integrals Consider the following regions R. a. Sketch the region R. b. Evaluate 4R dA to determine the area of the region. c. Evaluate 4R xy dA. R is the region bounded by the ellipse x2>18 + y2>36 = 1 with y 4x>3.

9396. Improper integrals Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 7.8). For example, under suitable conditions on f, L  a L h1x2 g1x2 f 1x, y2 dy dx = lim bS L b a L h1x2 g1x2 f 1x, y2 dy dx. Use or extend the one-variable methods for improper integrals to evaluate the following integrals. L _ 1 L e-x 0 xy dy dx

9396. Improper integrals Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 7.8). For example, under suitable conditions on f, L  a L h1x2 g1x2 f 1x, y2 dy dx = lim bS L b a L h1x2 g1x2 f 1x, y2 dy dx. Use or extend the one-variable methods for improper integrals to evaluate the following integrals. L _ 1 L 1>x2 0 2y x dy dx

9396. Improper integrals Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 7.8). For example, under suitable conditions on f, L  a L h1x2 g1x2 f 1x, y2 dy dx = lim bS L b a L h1x2 g1x2 f 1x, y2 dy dx. Use or extend the one-variable methods for improper integrals to evaluate the following integrals. L _ 0 L _ 0 e-x - y dy dx

9396. Improper integrals Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 7.8). For example, under suitable conditions on f, L  a L h1x2 g1x2 f 1x, y2 dy dx = lim bS L b a L h1x2 g1x2 f 1x, y2 dy dx. Use or extend the one-variable methods for improper integrals to evaluate the following integrals. L _ - _ L _ - _ 1 1x2 + 121y2 + 12 dy dx

97101. Volumes Compute the volume of the following solids. Sliced block The solid bounded by the planes x = 0, x = 5, z = y - 1, z = -2y - 1, z = 0, and z = 2

97101. Volumes Compute the volume of the following solids. Tetrahedron A tetrahedron with vertices 10, 0, 02, 1a, 0, 02, 1b, c, 02, and 10, 0, d2, where a, b, c, and d are positive real numbers

97101. Volumes Compute the volume of the following solids. Square column The column with a square base R = 51x, y2: _ x _ 1, _ y _ 16 cut by the plane z = 4 - x - y

97101. Volumes Compute the volume of the following solids. Wedge The wedge sliced from the cylinder x2 + y2 = 1 by the planes z = 1 - x and z = x - 1

97101. Volumes Compute the volume of the following solids. Wedge The wedge sliced from the cylinder x2 + y2 = 1 by the planes z = a12 - x2 and z = a1x - 22, where a 7 0

Existence of improper double integral For what values of m and n does the integral L _ 1 L 1>x 0 ym xn dy dx have a finite value? See Exercises 9396.

Existence of improper double integral Let R1 = 51x, y2: x 1, 1 y 26 and R2 = 51x, y2: 1 x 2, y 16. For n 7 1, which integral(s) have finite values: 4R1 x -n dA or 4R2 x -n dA?