Chapter 13: Q. 57 (page 1028)

Sketch the region of integration for each of integrals in Exercises $57 - 60$ , and then evaluate the integral by converting to polar coordinates.
$\int_{0}^{3 \sqrt{2} / 2} \int_{x}^{\sqrt{9 - x^{2}}} d y d x$

Short Answer

Expert verified

The value of the integral is

$\int_{0}^{3 \sqrt{2} / 2} \int_{x}^{\sqrt{9 - x^{2}}} d y d x = \frac{9 π}{8}$

Step by step solution

Given information

The integral is

$I = \int_{0}^{3 \sqrt{2} / 2} \int_{x}^{\sqrt{9 - x^{2}}} d y d x$

Here, $x = 0$ and $x = \frac{3 \sqrt{2}}{2}, y = x$ and $y = \sqrt{9 - x^{2}}$

Calculation

The region of integration $R$ is shown in the figure

Substitute $x = r \cos θ$ and $y = r \sin θ$ in the lower limit of $y$ .

$r \sin θ = r \cos θ \tan θ = 1 \Rightarrow θ = \frac{π}{4}$

Substitute $x = r \cos θ$ and $y = r \sin θ$ in the upper limit of $y$ .

$r \sin θ = \sqrt{9 - r^{2} \cos^{2} θ} r^{2} \sin^{2} θ = 9 - r^{2} \cos^{2} θ r^{2} \sin^{2} θ + r^{2} \cos^{2} θ = 9 r^{2} = 9 \Rightarrow r = \pm 3$

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

$r \cos θ = 0 \Rightarrow r = 0, θ = \frac{π}{2}$

Thus, the limits of $r$ are $r = 0$ and $r = 3$ and that of $θ$ are $\frac{π}{4}$ and $\frac{π}{2}$ .

$d x d y = r d r d θ$

Therefore,

$I = \int_{0}^{3 \sqrt{2} / 2} \int_{x}^{\sqrt{9 - x^{2}}} d y d x = \int_{π / 4}^{π / 2} \int_{0}^{3} r d r d θ I = \int_{π / 4}^{π / 2} [\int_{0}^{3} r d r] θ$

Integrate with respect to $r$ first

$I = \int_{s / 4}^{π / 2} {[\frac{r^{2}}{2}]}_{0}^{3} d θ I = \int_{π / 4}^{π / 2} [\frac{9}{2}] d θ I = \frac{9}{2} [θ]_{n / 4}^{π / 2} I = \frac{9}{2} [\frac{π}{2} - \frac{π}{4}] I = \frac{9 π}{8}$

Thus, the value of the integral is localid="1651390111195" $\int_{0}^{3 \sqrt{2} / 2} \int_{x}^{\sqrt{9 - x^{2}}} d y d x = \frac{9 π}{8}$

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Sketch the region of integration for each of integrals in Exercises $57 - 60$ , and then evaluate the integral by converting to polar coordinates.
$\int_{0}^{3 \sqrt{2} / 2} \int_{x}^{\sqrt{9 - x^{2}}} d y d x$

Short Answer

Step by step solution

Given information

Calculation

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Sketch the region of integration for each of integrals in Exercises57-60, and then evaluate the integral by converting to polar coordinates.∫032/2∫x9-x2dydx

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

Calculus

Discrete Mathematics

Probability and Statistics

Pure Maths

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

Sketch the region of integration for each of integrals in Exercises $57 - 60$ , and then evaluate the integral by converting to polar coordinates.
$\int_{0}^{3 \sqrt{2} / 2} \int_{x}^{\sqrt{9 - x^{2}}} d y d x$