Chapter 13: Q. 23 (page 1082)

Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.
$\int_{- 3}^{3} \int_{- \sqrt{9 - x^{2}}}^{\sqrt{9 - x^{2}}} \frac{x + 2 y}{x^{2} + y^{2}} d y d x$

Short Answer

Expert verified

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

Step by step solution

Step !: Draw the region

From the limits of integration, the region is shown below,

Convert into polar form

By using the below substitution,

$x = r \cos θ y = r \sin θ x^{2} + y^{2} = r^{2} d x d y = r d r d θ$

The equivalent polar integral of the given integral is,

$\int_{- 3}^{3} \int_{- \sqrt{9 - x^{2}}}^{\sqrt{9 - x^{2}}} \frac{x + 2 y}{x^{2} + y^{2}} d y d x \Rightarrow \int_{0}^{2 π} \int_{0}^{3} (\cos θ + 2 \sin θ) d r d θ$

Calculate the integral

$I = \int_{0}^{2 π} \int_{0}^{3} (\cos θ + 2 \sin θ) d r d θ I = \int_{0}^{2 π} (\cos θ + 2 \sin θ) d θ \int_{0}^{3} r d r I = {[\sin θ - 2 \cos θ]}_{0}^{2 π} \int_{0}^{3} r d r I = [- 2 - (- 2)] \int_{0}^{3} r d r I = (- 2 + 2) \int_{0}^{3} r d r I = 0$

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Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.
$\int_{- 3}^{3} \int_{- \sqrt{9 - x^{2}}}^{\sqrt{9 - x^{2}}} \frac{x + 2 y}{x^{2} + y^{2}} d y d x$

Short Answer

Step by step solution

Step !: Draw the region

Convert into polar form

Calculate the integral

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Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.∫-33∫-9-x29-x2x+2yx2+y2dydx

Short Answer

Step by step solution

Step !: Draw the region

Convert into polar form

Calculate the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Decision Maths

Pure Maths

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.
$\int_{- 3}^{3} \int_{- \sqrt{9 - x^{2}}}^{\sqrt{9 - x^{2}}} \frac{x + 2 y}{x^{2} + y^{2}} d y d x$