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Chapter 13: Q. 4 (page 1082)

Evaluating a double integral as an iterated integral: Use Fubini鈥檚 theorem to evaluate the given double integrals. For each integral, show that you obtain the same result when you integrate using both possible orders of integration when R = {(x, y) | 0 鈮� x 鈮� 2 and 1 鈮� y 鈮� 4}.
$鈭� (x y^{2}) d A$

Short Answer

Expert verified

$鈭� (x y^{2}) d A = 42 s q u a r e u n i t s$

Step by step solution

01

Fubini’s theorem

Let a< b and c<d be real numbers, let R be the rectangle defined by, $R = \{(x, y) | a 鈮� x 鈮� b a n d c 鈮� y 鈮� d\}$ and let f(x) is continuous on R, Then

$鈭� f (x, y) d A = {鈭�}_{a}^{b} {鈭�}_{c}^{d} f (x, y) d x d y$

02

Evaluate the integral

$鈭� (x y^{2}) d A = {鈭�}_{1}^{4} {鈭�}_{0}^{2} (x y^{2}) d x d y = {鈭�}_{1}^{4} [{鈭�}_{0}^{2} (x y^{2}) d x] d y = {鈭�}_{1}^{4} d y {[\frac{x^{2}}{2} y^{2}]}_{0}^{2} = {鈭�}_{1}^{4} d y [\frac{2^{2}}{2} y^{2}] = {鈭�}_{1}^{4} (2 y^{2}) d y = {[2 \frac{y^{3}}{3}]}_{1}^{4} = \frac{128}{3} - \frac{2}{3} = \frac{126}{3} = 42 s q u a r e u n i t s$

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Most popular questions from this chapter

Let $f (x, y, z)$ be a continuous function of three variables, let localid="1650352548375" $惟_{y z} = {(y, z) | a 鈮� y 鈮� b a n d h_{1} (y) 鈮� z 鈮� h_{2} (y)}$ be a set of points in the $y z$ -plane, and let localid="1650354983053" $惟 = {(x, y, z) | (y, z) 鈭� 惟_{y z} a n d g_{1} (y, z) 鈮� x 鈮� g_{2} (y, z)}$ be a set of points in $3$ -space. Find an iterated triple integral equal to the triple integral localid="1650353288865" $\underset{惟}{鈭�} f (x, y, z) d V$ . How would your answer change iflocalid="1650352747263" $惟_{y z} = {(y, z) | a 鈮� z 鈮� b a n d h_{1} (z) 鈮� x 鈮� h_{2} (z)}$ ?

Evaluate the sums in Exercises $23 鈥� 28$ .

{鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} (j - k)

Evaluate the iterated integral :

${鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3} 鈥� {鈭�}_{0}^{\sin^{鈭� 1} 鈦� x} 鈥� (x + y) d z d y d x$

Find the masses of the solids described in Exercises 53鈥�56.

The first-octant solid bounded by the coordinate planes and the plane 3x + 4y + 6z = 12 if the density at each point is proportional to the distance of the point from the xz-plane.

Use the results of Exercises 59 and 60 to find the centers of masses of the lamin忙 in Exercises 61鈥�67.

Use the lamina from Exercise 61, but assume that the density is proportional to the distance from the x-axis.

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