Q. 59 Sketch the region of integration... [FREE SOLUTION]

Chapter 13: Q. 59 (page 1028)

Sketch the region of integration for each of integrals in Exercises $57 鈥� 60$ , and then evaluate the integral by converting to polar coordinates.
${鈭�}_{0}^{4} {鈭�}_{0}^{\sqrt{16 - x^{2}}} e^{x^{2}} e^{y^{2}} d y d x$

Short Answer

Expert verified

The value of the integral is ${鈭�}_{0}^{4} {鈭�}_{0}^{\sqrt{16 - x^{2}}} e^{x^{2}} e^{y^{2}} d y d x = \frac{蟺}{4} (e^{16} - 1)$

Step by step solution

Given information

The integral is $I = {鈭�}_{0}^{4} {鈭�}_{0}^{\sqrt{16 - x^{2}}} e^{x^{2}} e^{y^{2}} d y d x$

Here, $x = 0$ and $x = 4, y = 0$ and $y = \sqrt{16 - x^{2}}$

Calculation

The region of integration R is shown in the figure

$r^{2} \sin^{2} 胃 + r^{2} \cos^{2} 胃 = 16 r^{2} = 16 鈬� r = 4$

Substitute $x = r \cos 胃$ in the lower limit of $x$ .

$r \cos 胃 = 0 鈬� r = 0, 胃 = \frac{蟺}{2}$

Thus, the limits of $r$ are $r = 0$ and $r = 4$ and that of $胃$ are $0$ and $\frac{蟺}{2} .$

$d x d y = r d r d 胃$

Therefore,

$I = {鈭�}_{0}^{4} {鈭�}_{0}^{\sqrt{16 - x^{2}}} e^{x^{2}} e^{y^{2}} d y d x = {鈭�}_{0}^{x / 2} {鈭�}_{0}^{4} e^{r^{2}} r d r d 胃 I = {鈭�}_{0}^{蟺 / 2} [{鈭�}_{0}^{4} r e^{r^{2}} d r] d 胃$

Integrate with respect to $r$ first

Put $r^{2} = t$

$2 r d r = d t 鈬� r d r = \frac{d t}{2} I = {鈭�}_{0}^{蟺 / 2} [{鈭�}_{0}^{16} e^{t} \frac{d t}{2}] d 胃 I = \frac{1}{2} {鈭�}_{0}^{蟺 / 2} {[e^{t}]}_{0}^{16} d 胃 I = \frac{1}{2} {鈭�}_{0}^{蟺 / 2} [e^{16} - e^{0}] d 胃 I = \frac{1}{2} {鈭�}_{0}^{蟺 / 2} [e^{16} - 1] d 胃 I = \frac{1}{2} (e^{16} - 1) [胃]_{0}^{蟺 / 2} I = \frac{1}{2} (e^{16} - 1) [\frac{蟺}{2}] I = \frac{蟺}{4} (e^{16} - 1)$

Thus, the value of the integral is

${鈭�}_{0}^{4} {鈭�}_{0}^{\sqrt{16 - x^{2}}} e^{x^{2}} e^{y^{2}} d y d x = \frac{蟺}{4} (e^{16} - 1)$

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91影视

Sketch the region of integration for each of integrals in Exercises $57 鈥� 60$ , and then evaluate the integral by converting to polar coordinates.
${鈭�}_{0}^{4} {鈭�}_{0}^{\sqrt{16 - x^{2}}} e^{x^{2}} e^{y^{2}} d y d x$

Short Answer

Step by step solution

Given information

Calculation

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91影视

Sketch the region of integration for each of integrals in Exercises 57鈥�60, and then evaluate the integral by converting to polar coordinates.鈭�04鈭�016-x2ex2ey2dydx

Short Answer

Step by step solution

Given information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Probability and Statistics

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Sketch the region of integration for each of integrals in Exercises $57 鈥� 60$ , and then evaluate the integral by converting to polar coordinates.
${鈭�}_{0}^{4} {鈭�}_{0}^{\sqrt{16 - x^{2}}} e^{x^{2}} e^{y^{2}} d y d x$