Chapter 13: Q 62. (page 1014)

Evaluate the double integral over the specified region.
$\iint_{Ω} d A,$ where $Ω$ is region in first quadrant bounded by curves $y = x^{m} and y = x^{n}$ where $m, n$ are distinct positive integers.

Short Answer

Expert verified

The required integral is $\frac{| m - n |}{(m + 1) (n + 1)}$ .

Step by step solution

Given Information

We are given double integral as $\iint_{Ω} d A$ , region is bounded by curves $y = x^{m} and x = y^{n}$

Considering type I integral

The curves will intersect at $x = 1$

For type I integral, $0 \leq x \leq 1, x^{m} \leq y \leq x^{n}$

Simplification

The integral is written as:

$\iint_{Ω} d A = \int_{0}^{1} \int_{x^{''}}^{x^{n}} d y d x$

$\int_{0}^{1} \int_{x^{n}}^{x^{n}} d y d x = \int_{0}^{1} [y]_{x^{m}}^{x^{n}} d x$

$\int_{0}^{1} \int_{x^{m}}^{x^{n}} d y d x = {[\frac{x^{n + 1}}{n + 1} - \frac{x^{m + 1}}{m + 1}]}_{0}^{1}$

$= [\frac{1}{n + 1} - \frac{1}{m + 1}]$

$= \frac{m - n}{(m + 1) (n + 1)}$

Integral is $\frac{| m - n |}{(m + 1) (n + 1)}$

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Evaluate the double integral over the specified region.
$\iint_{Ω} d A,$ where $Ω$ is region in first quadrant bounded by curves $y = x^{m} and y = x^{n}$ where $m, n$ are distinct positive integers.

Short Answer

Step by step solution

Given Information

Considering type I integral

Simplification

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Evaluate the double integral over the specified region.∬ΩdA,where Ωis region in first quadrant bounded by curves y=xmandy=xnwherem,nare distinct positive integers.

Short Answer

Step by step solution

Given Information

Considering type I integral

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Evaluate the double integral over the specified region.
$\iint_{Ω} d A,$ where $Ω$ is region in first quadrant bounded by curves $y = x^{m} and y = x^{n}$ where $m, n$ are distinct positive integers.