Answered: Evaluate the integral of the function…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Evaluate the integral of the function f(x, y) over the region R that is bounded by the x-axis, the line x = 4, and the curve y = sqrt(x). Set up integrals with both dA = dx dy and dA = dy dx.

Expert Solution

Step 1: Analysis and Introduction

Given information:

The region $R$ is bounded by $x$ axis, the line $x equals 4$ and the curve $y equals square root of x$ .

Any function $f open parentheses x comma y close parentheses$ is defined over the region $R$ .

Objective:

Set up the integral of the function $f open parentheses x comma y close parentheses$ over the region $R$ in the form $d x d y$ and $d y d x$ .

Concept used:

The double integral of the function $f open parentheses x comma y close parentheses$ over the region $R$ is determined as:

$double integral subscript R f open parentheses x comma y close parentheses space d A equals integral subscript y equals c end subscript superscript y equals d end superscript integral subscript x equals g subscript 1 open parentheses y close parentheses end subscript superscript x equals g subscript 2 open parentheses y close parentheses end superscript f open parentheses x comma y close parentheses space d x space d y equals integral subscript x equals a end subscript superscript x equals b end superscript integral subscript y equals h subscript 1 open parentheses x close parentheses end subscript superscript y equals h subscript 2 open parentheses x close parentheses end superscript f open parentheses x comma y close parentheses space d y space d x$