Chapter 13: Q. 10 (page 1082)

Evaluating iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.
$\int_{0}^{2} \int_{0}^{y} x \sqrt{y^{2} - x^{2}} d x d y$

Short Answer

Expert verified

$\int_{0}^{2} \int_{0}^{y} x \sqrt{y^{2} - x^{2}} d x d y = \frac{128}{21}$

Step by step solution

Draw the region

The region determined by the limits of the given iterated integral is shown below,

Evaluate the given integral

$I = \int_{0}^{y} x \sqrt{y^{2} - x^{2}} d x Substitute, y^{2} - x^{2} = u^{2} Differentiate w . r . t . x - 2 x d x = 2 u d u d x = - \frac{u}{x} d u$

$I = - \int_{0}^{y} x \sqrt{u^{2}} \frac{u}{x} d u I = - \int_{0}^{y} u^{2} d u I = - [\frac{u^{3}}{3}] I = - {[\frac{{(y^{2} - x^{2})}^{3}}{3}]}_{0}^{y} I = \frac{y^{6}}{3}$

Therefore,

$\int_{0}^{2} [\int_{0}^{y} x \sqrt{y^{2} - x^{2}} d x] d y = \frac{1}{3} \int_{0}^{2} y^{6} d y = \frac{1}{3} {[\frac{y^{7}}{7}]}_{0}^{2} = \frac{128}{21}$

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Evaluating iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.
$\int_{0}^{2} \int_{0}^{y} x \sqrt{y^{2} - x^{2}} d x d y$

Short Answer

Step by step solution

Draw the region

Evaluate the given integral

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Evaluating iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.∫02∫0yxy2-x2dxdy

Short Answer

Step by step solution

Draw the region

Evaluate the given integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Evaluating iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.
$\int_{0}^{2} \int_{0}^{y} x \sqrt{y^{2} - x^{2}} d x d y$