Let f (y) be a function of y alone and set G(t) = t 0 x 0 f (y) dy dx. (a) Use the

Which of the following expressions do not make sense? (a) 1 0 x 1 f (x, y) dy dx (b) 1 0 y 1 f (x, y) dy dx (c) 1 0 y x f (x, y) dy dx (d) 1 0 1 x f (x, y) dy dx 2. Draw

Draw a domain in the plane that is neither vertically nor horizontally simple.

Which of the four regions in Figure 20 is the domain of integration for 0 2/2 1x2 x f (x, y) dy dx? x

Let D be the unit disk. If the maximum value of f (x, y) on D is 4, then the largest possible value of D f (x, y) dA is (choose the correct answer): a)4 b)4r c)4/r

In Exercises 57, compute the double integral of f (x, y) = x2y over the given shaded domain in Figure 24. 5. (A)

In Exercises 57, compute the double integral of f (x, y) = x2y over the given shaded domain in Figure 24.(B)

In Exercises 57, compute the double integral of f (x, y) = x2y over the given shaded domain in Figure 24.(C)

Sketch the domain D defined by x + y 12, x 4, y 4 and compute D ex+y dA. 9

Integrate f (x, y) = x over the region bounded by y = x2 and y = x + 2.

Sketch the region D between y = x2 and y = x(1 x). Express D as a simple region and calculate the integral of f (x, y) = 2y over D.

Evaluate D y x dA, where D is the shaded part of the semicircle of radius 2 in Figure 25. 1

Calculate the double integral of f (x, y) = y2 over the rhombus R in Figure 26.

Calculate the double integral of f (x, y) = x + y over the domain D = {(x, y) : x2 + y2 4, y 0}.

Integrate f (x, y) = (x + y + 1)2 over the triangle with vertices (0, 0), (4, 0), and (0, 8).

Calculate the integral of f (x, y) = x over the region D bounded above by y = x(2 x) and below by x = y(2 y). Hint: Apply the quadratic formula to the lower boundary curve to solve for y as a function of x.

Integrate f (x, y) = x over the region bounded by y = x, y = 4x x2, and y = 0 in two ways: as a vertically simple region and as a horizontally simple region.

In Exercises 1724, compute the double integral of f (x, y) over the domain D indicated. f (x, y) = x2y; 1 x 3, x y 2x + 1

In Exercises 1724, compute the double integral of f (x, y) over the domain D indicated. f (x, y) = 1; 0 x 1, 1 y ex

In Exercises 1724, compute the double integral of f (x, y) over the domain D indicated. f (x, y) = x; 0 x 1, 1 y ex2

In Exercises 1724, compute the double integral of f (x, y) over the domain D indicated. f (x, y) = cos(2x + y); 1 2 x 2 , 1 y 2x

In Exercises 1724, compute the double integral of f (x, y) over the domain D indicated. f (x, y) = 2xy; bounded by x = y, x = y2

In Exercises 1724, compute the double integral of f (x, y) over the domain D indicated. f (x, y) = sin x; bounded by x = 0, x = 1, y = cos x

In Exercises 1724, compute the double integral of f (x, y) over the domain D indicated. f (x, y) = ex+y ; bounded by y = x 1, y = 12 x for 2 y 4

In Exercises 1724, compute the double integral of f (x, y) over the domain D indicated. f (x, y) = (x + y)1; bounded by y = x, y =1,y=e,x=0

In Exercises 2528, sketch the domain of integration and express as an iterated integral in the opposite order. 25. 4 0 4 x f (x, y) dy dx 2

In Exercises 2528, sketch the domain of integration and express as an iterated integral in the opposite order9 4 3 y f (x, y) dx dy 2

In Exercises 2528, sketch the domain of integration and express as an iterated integral in the opposite order9 4 y 2 f (x, y) dx dy 28

In Exercises 2528, sketch the domain of integration and express as an iterated integral in the opposite order1 0 e ex f (x, y) dy dx 2

Sketch the domain D corresponding to 4 0 2 y  4x2 + 5y dx dy Then change the order of integration and evaluate. 30

Change the order of integration and evaluate 1 0 /2 0 x cos(xy) dx dy Explain the simplification achieved by changing the order.

Compute the integral of f (x, y) = (ln y)1 over the domain D bounded by y = ex and y = e x . Hint: Choose the order of integration that enables you to evaluate the integral.

Evaluate by changing the order of integration: 4 0 2 x sin y3 dy dx I

In Exercises 3336, sketch the domain of integration. Then change the order of integration and evaluate. Explain the simplification achieved by changing the order. 33. 1 0 1 y sin x x dx dy 34

In Exercises 3336, sketch the domain of integration. Then change the order of integration and evaluate. Explain the simplification achieved by changing the order.4 0 2 y  x3 + 1 dx dy 3

In Exercises 3336, sketch the domain of integration. Then change the order of integration and evaluate. Explain the simplification achieved by changing the order.1 0 1 y=x xey3 dy dx 36.

In Exercises 3336, sketch the domain of integration. Then change the order of integration and evaluate. Explain the simplification achieved by changing the order.1 0 1 y=x2/3 xey4 dy dx 3

Sketch the domain D where 0 x 2, 0 y 2, and x or y is greater than 1. Then compute D ex+y dA. 38

Calculate D ex dA, where D is bounded by the lines y = x + 1, y = x, x = 0, and x = 1. I

In Exercises 3942, calculate the double integral of f (x, y) over the triangle indicated in Figure 27.f (x, y) = ex2 , (A)

In Exercises 3942, calculate the double integral of f (x, y) over the triangle indicated in Figure 27.f (x, y) = 1 2x, (B)

In Exercises 3942, calculate the double integral of f (x, y) over the triangle indicated in Figure 27.f (x, y) = x y2 , (C)

In Exercises 3942, calculate the double integral of f (x, y) over the triangle indicated in Figure 27.f (x, y) = x + 1, (D)

Calculate the double integral of f (x, y) = sin y y over the region D in Figure 28.

Evaluate D x dA for D in Figure 29.

Find the volume of the region bounded by z = 40 10y, z = 0, y = 0, and y = 4 x2.

Find the volume of the region enclosed by z = 1 y2 and z = y2 1 for 0 x 2.

Find the volume of the region bounded by z = 16 y, z = y, y = x2, and y = 8 x2.

Find the volume of the region bounded by y = 1 x2, z = 1, y = 0, and z + y = 2.

Set up a double integral that gives the volume of the region bounded by the two paraboloids z = x2 + y2 and z = 8 x2 y2. (Do not evaluate the double integral.)

Set up a double integral that gives the volume of the region bounded by z = 1 y2, z = y, x = 0, y = 0, and x + y = 1. (Do not evaluate the double integral.)

Calculate the average value of f (x, y) = ex+y on the square [0, 1][0, 1].

Calculate the average height above the x-axis of a point in the

Find the average height of the ceiling in Figure 30 defined by z = y2 sin x for 0 x , 0 y 1.

Calculate the average value of the x-coordinate of a point on the semicircle x2 + y2 R2, x 0. What is the average value of the ycoordinate?

What is the average value of the linear function f (x, y) = mx + ny + p on the ellipse x a 2 + y b 2 1? Argue by symmetry rather than calculation

Find the average square distance from the origin to a point in the domain D in Figure 31.

Let D be the rectangle 0 x 2, 1 8 y 1 8 , and let f (x, y) =  x3 + 1. Prove that D f (x, y) dA 3 2

(a) Use the inequality 0 sin x x for x 0 to show that 1 0 1 0 sin(xy) dx dy 1 4 (b) Use a computer algebra system to evaluate the double integral to three decimal places. 5

Prove the inequality D dA 4 + x2 + y2 , where D is the disk x2 + y2 4. 6

Let D be the domain bounded by y = x2 + 1 and y = 2. Prove the inequality 4 3 D (x2 + y2)dA 20 3 6

Let f be the average of f (x, y) = xy2 on D = [0, 1][0, 4]. Find a point P D such that f (P ) = f (the existence of such a point is guaranteed by the Mean Value Theorem for Double Integrals).

Verify the Mean Value Theorem for Double Integrals for f (x, y) = exy on the triangle bounded by y = 0, x = 1, and y = x.

In Exercises 63 and 64, use (11) to estimate the double integral. 63. The following table lists the areas of the subdomains Dj of the domain D in Figure 32 and the values of a function f (x, y) at sample points Pj Dj . Estimate D f (x, y) dA. j 123456 Area(Dj ) 1.2 1.1 1.4 0.6 1.2 0.8 f (Pj ) 9 9.1 9.3 9.1 8.9 8.8 D

The domain D between the circles of radii 5 and 5.2 in the first quadrant in Figure 33 is divided into six subdomains of angular width = 12 , and the values of a function f (x, y) at sample points are given. Compute the area of the subdomains and estimate D f (x, y) dA. x

According to Eq. (3), the area of a domain D is equal to D 1 dA. Prove that if D is the region between two curves y = g1(x) and y = g2(x) with g2(x) g1(x) for a x b, then

Let D be a closed connected domain and let P,Q D. The Intermediate Value Theorem (IVT) states that if f is continuous on D, then f (x, y) takes on every value between f (P ) and f (Q) at some point in D. (a) Show, by constructing a counterexample, that the IVT is false if D is not connected. (b) Prove the IVT as follows: Let c(t) be a path such that c(0) = P and c(1) = Q (such a path exists because D is connected). Apply the IVT in one variable to the composite function f (c(t)).

Use the fact that a continuous function on a closed domain D attains both a minimum value m and a maximum value M, together with Theorem 3, to prove that the average value f lies between m and M. Then use the IVT in Exercise 62 to prove the Mean Value Theorem for Double Integrals.

Let f (y) be a function of y alone and set G(t) = t 0 x 0 f (y) dy dx. (a) Use the Fundamental Theorem of Calculus to prove that G (t) = f (t). (b) Show, by changing the order in the double integral, that G(t) = t 0 (t y)f (y) dy. This illustrates that the second antiderivative of f (y) can be expressed as a single integral.