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Chapter 13: Q 60. (page 1016)

Evaluate the double integral over the specified region.
${鈭�}_{惟} x e^{x^{3}} d A$ where $惟$ is the triangular region in the first quadrant bounded below by the x-axis, bounded above $y = m x_{,} where m > 0$ and bounded on the
right by the line with equation $x = 1$ .

Short Answer

Expert verified

The required integral is $\frac{m}{3} (e - 1)$ .

Step by step solution

01

Given Information

We are given following double integral as ${鈭�}_{惟} x e^{x^{3}} d A$

$惟$ is triangular region in first quadrant bounded by $y = m x$ , slope $m > 0$

bounded on right by $x = 1$

02

Simplification

Region of integration is region between line $y = m x, y = 0, x = 1$

It can be seen that for type I integral, $0 鈮� x 鈮� 1, 0 鈮� y 鈮� m x$

Therefore

${鈭�}_{惟} x e^{x^{3}} d A = {鈭�}_{0}^{1} {鈭�}_{0}^{m x} x e^{x^{3}} d y d x$

${鈭�}_{0}^{1} {鈭�}_{0}^{m x} x e^{x^{3}} d y d x = {鈭�}_{0}^{1} [y]_{0}^{m x} x e^{x^{3}} d x$

$= m {鈭�}_{0}^{1} x^{2} e^{x^{3}} d x$

Using $x^{3} = t$

$鈬� 3 x^{2} d x = d t$

Limits are

$t = 0^{3} = 0, t = 1^{3} = 1$

${鈭�}_{0}^{1} {鈭�}_{0}^{m x} x e^{x^{3}} d y d x = \frac{m}{3} {鈭�}_{0}^{1} e^{t} d t$

$= \frac{m}{3} {[e^{t}]}_{0}^{1}$

$= \frac{m}{3} (e - 1)$

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Most popular questions from this chapter

In the following lamina, all angles are right angles and the density is constant:

Describe the three-dimensional region expressed in each iterated integral:

${鈭�}_{鈭� 2}^{4} 鈥� {鈭�}_{2}^{6} 鈥� {鈭�}_{0}^{5} 鈥� f (x, y, z) d y d x d z$

$How many summands are in {鈭�}_{j = 3}^{13} {鈭�}_{k = 5}^{20} \frac{j^{2}}{e^{jk}} ?$

Identify the quantities determined by the integral expressions in Exercises 19鈥�24. If x, y, and z are all measured in centimeters and 蚁(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

${鈭�}_{惟} 鈥� x 蚁 (x, y, z) d V$

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

${鈭�}_{0}^{3} {鈭�}_{0}^{3} {鈭�}_{0}^{3 - y} f (x, y, z) d z d y d x$

See all solutions

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