Chapter 13: Q 61. (page 1014)

Evaluate the double integral over the specified region.
$\iint_{Ω} x^{2} y^{3} d A$ where $Ω$ is region in first quadrant bounded by curves $y = x^{2} and x = y^{2}$

Short Answer

Expert verified

The integral is $\frac{3}{110}$ .

Step by step solution

Given Information

We are given with $\iint_{Ω} x^{2} y^{3} d A$

The curves are $y = x^{2} and x = y^{2}$ .

Graph

The region in graph is shown below:

Simplification

The region of integration is region between $y = x^{2} and y = \sqrt{x}$ .

For type I integral

$0 \leq x \leq 1, x^{2} \leq y \leq \sqrt{x}$

Hence, integral is simplified as

$\int_{0}^{1} \int_{x^{2}}^{\sqrt{x}} x^{2} y^{3} d y d x = \int_{0}^{1} {[\frac{y^{4}}{4}]}_{x^{2}}^{\sqrt{x}} x^{2} d x$

$= \frac{1}{4} \int_{0}^{1} (x^{2} - x^{8}) x^{2} d x$

$= \frac{1}{4} \int_{0}^{1} (x^{4} - x^{10}) d x$

$= \frac{1}{4} {[\frac{x^{5}}{5} - \frac{x^{11}}{11}]}_{0}^{1}$

$= \frac{1}{4} [\frac{1}{5} - \frac{1}{11}]$

$= \frac{3}{110}$

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Evaluate the double integral over the specified region.
$\iint_{Ω} x^{2} y^{3} d A$ where $Ω$ is region in first quadrant bounded by curves $y = x^{2} and x = y^{2}$

Short Answer

Step by step solution

Given Information

Graph

Simplification

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Evaluate the double integral over the specified region.∬Ωx2y3dAwhere Ω is region in first quadrant bounded by curvesy=x2andx=y2

Short Answer

Step by step solution

Given Information

Graph

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Evaluate the double integral over the specified region.
$\iint_{Ω} x^{2} y^{3} d A$ where $Ω$ is region in first quadrant bounded by curves $y = x^{2} and x = y^{2}$