Chapter 13: Q. 16 (page 1082)

Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.
$\int_{0}^{1} \int_{\sqrt{x}}^{1} e^{\frac{x}{y^{2}}} d y d x$

Short Answer

Expert verified

$\int_{0}^{1} \int_{\sqrt{x}}^{1} e^{\frac{x}{y^{2}}} d y d x = \frac{1}{3} (e - 1)$

Step by step solution

Draw the region

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

Reversing the order of integration

From the diagram, the order of differentiation is changed,

$\int_{0}^{1} \int_{\sqrt{x}}^{1} e^{\frac{x}{y^{2}}} d y d x \Rightarrow \int_{0}^{1} \int_{0}^{y^{2}} e^{\frac{x}{y^{2}}} d y d x$

Evaluate the integral

$I = \int_{0}^{1} [\int_{0}^{y^{2}} e^{\frac{x}{y^{2}}} d x] d y I = \int_{0}^{1} {[\frac{e^{\frac{x}{y^{2}}}}{1 / y^{2}}]}_{0}^{y^{2}} d y I = \int_{0}^{1} y^{2} [e - 1] d y I = (e - 1) {[\frac{y^{3}}{3}]}_{0}^{1} I = \frac{1}{3} (e - 1)$

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Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.
$\int_{0}^{1} \int_{\sqrt{x}}^{1} e^{\frac{x}{y^{2}}} d y d x$

Short Answer

Step by step solution

Draw the region

Reversing the order of integration

Evaluate the integral

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Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.∫01∫x1exy2dydx

Short Answer

Step by step solution

Draw the region

Reversing the order of integration

Evaluate the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Decision Maths

Calculus

Mechanics Maths

Geometry

Study anywhere. Anytime. Across all devices.

Reversing the order of integration: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals by reversing the order of integration.
$\int_{0}^{1} \int_{\sqrt{x}}^{1} e^{\frac{x}{y^{2}}} d y d x$