Chapter 13: Q. 19 (page 1082)

Using polar coordinates to evaluate iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.
$\int_{0}^{π / 2} \int_{0}^{2 \sin θ} r^{3} d r d θ$

Short Answer

Expert verified

$\int_{0}^{π / 2} \int_{0}^{2 \sin θ} r^{3} d r d θ = \frac{3}{4} π$

Step by step solution

Draw the region

The region determined by the limits is shown below,

Evaluate the integral

$I = \int_{0}^{π / 2} \int_{0}^{2 \sin θ} r^{3} d r d θ I = \int_{0}^{π / 2} {[\frac{r^{4}}{4}]}_{0}^{2 \sin θ} d θ I = \frac{16}{4} \int_{0}^{π / 2} \sin^{4} θ d θ I = 4 \int_{0}^{π / 2} \sin^{4} θ d θ I = 4 [\frac{3}{4} \cdot \frac{1}{2} \cdot \frac{π}{2}] I = \frac{3}{4} π$

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Using polar coordinates to evaluate iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.
$\int_{0}^{π / 2} \int_{0}^{2 \sin θ} r^{3} d r d θ$

Short Answer

Step by step solution

Draw the region

Evaluate the integral

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Using polar coordinates to evaluate iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.∫0π/2∫02sinθr3drdθ

Short Answer

Step by step solution

Draw the region

Evaluate the integral

One App. One Place for Learning.

Most popular questions from this chapter

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Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Using polar coordinates to evaluate iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.
$\int_{0}^{π / 2} \int_{0}^{2 \sin θ} r^{3} d r d θ$