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Calculus
Chapter 13
Evaluating iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.

Chapter 13: Evaluating iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals. (page 1082)

Q. 7 ${鈭�}_{0}^{1} {鈭�}_{- \sqrt{1 - y^{2}}}^{\sqrt{1 + y^{2}}} \frac{x}{y + 1} d x d y$

Short Answer

Expert verified

${鈭�}_{0}^{1} {鈭�}_{- \sqrt{1 - y^{2}}}^{\sqrt{1 + y^{2}}} \frac{x}{y + 1} d x d y = 0$

Step by step solution

Draw the region

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

Evaluate integral

Here a concept is used,

${鈭�}_{- a}^{a} f (x) d x = 0, If f (x) is an odd function i . e . f (- x) = - f (x)$

${鈭�}_{0}^{1} {鈭�}_{- \sqrt{1 - y^{2}}}^{\sqrt{1 + y^{2}}} \frac{x}{y + 1} d x d y = {鈭�}_{0}^{1} \frac{1}{y + 1} [{鈭�}_{- \sqrt{1 - y^{2}}}^{\sqrt{1 + y^{2}}} x d x] d y = {鈭�}_{0}^{1} \frac{1}{y + 1} [0] d y [f (x) = x, is an odd function] = 0$

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Q. 7 ${鈭�}_{0}^{1} {鈭�}_{- \sqrt{1 - y^{2}}}^{\sqrt{1 + y^{2}}} \frac{x}{y + 1} d x d y$

Short Answer

Step by step solution

Draw the region

Evaluate integral

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Q. 7鈭�01鈭�-1-y21+y2xy+1dxdy

Short Answer

Step by step solution

Draw the region

Evaluate integral

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

Q. 7 ${鈭�}_{0}^{1} {鈭�}_{- \sqrt{1 - y^{2}}}^{\sqrt{1 + y^{2}}} \frac{x}{y + 1} d x d y$