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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Evaluating a Double Integral In Exercises 85–88, evaluate the double integral.

0 4 0 16 x 2 2 x d y d x

To determine

To calculate: The simplified value of the double integral 04016x22x dy dx.

Explanation

Given Information:

The provided double integral is 04016x22x dy dx.

Formula used:

A double integral is defined as an integral of an integral. There are two types of double integral with function of two variables as,

abg1(x)g2(x)f(x,y)dy dx=abg1(x)g2(x)[f(x,y)dy] dxabg1(x)g2(x)f(x,y)dx dy=abg1(x)g2(x)[f(x,y)dx] dy

The integration of function f(x)=xn,n1 is;

xndx=xn+1n+1+c

Calculation:

Consider the double integral 04016x22x dy dx.

It is double integral of type 1 where x is the outer variable.

First integrate with respect to y and apply fundamental theorem of calculus as,

04016x22x dy dx=04[2xy]016x2dx=042x16x2dx=23[(16x2)3/2]04

Now, integrate with respect to x by taking z=16x2 so;

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