Evaluate the following integrals. | Ch 7.3 - 12E

Problem 12E Chapter 7.3

Calculus: Early Transcendentals | 1st Edition

2901 Step-by-step solutions solved by professors and subject experts
Get 24/7 help from StudySoup virtual teaching assistants

Calculus: Early Transcendentals | 1st Edition

4 5 0 237 Reviews

Problem 12E

Evaluate the following integrals.

Step-by-Step Solution:

Solution:-

Step 1</p>

We have to integrate $\int\limits_{\ }^{\ }\sqrt{36-{x}^{2}}dx$

To evaluate this integral we use the method of integration by parts.

I = $\int\limits_{\ }^{\ }\sqrt{36-{x}^{2}}dx$

Multiplying and dividing the integrand by $\sqrt{36-{x}^{2}}$ ,we get

I = $\int\limits_{\ }^{\ }\sqrt{36-{x}^{2}}dx\ =\ \int\limits_{\ }^{\ }\frac{36-{x}^{2}}{\sqrt{36-{x}^{2}}\ }dx\$

= $36\int\limits_{\ }^{\ }\frac{dx}{\sqrt{36-{x}^{2}}}\ -\ \int\limits_{\ }^{\ }\frac{{x}^{2}dx}{\sqrt{36-{x}^{2}}}\$

Now, taking I1 = $\int\limits_{\ }^{\ }\frac{dx}{\sqrt{36-{x}^{2}}}$ and I2 = $\ \int\limits_{\ }^{\ }\frac{{x}^{2}dx}{\sqrt{36-{x}^{2}}}$

I = 36I1 $-$ I2

Step 2</p>

To solve for I1 = $\int\limits_{\ }^{\ }\frac{dx}{\sqrt{36-{x}^{2}}}$ ,we go ahead with the method of integration by substitution.

But first let’s arrange the integral,

$\int\limits_{\ }^{\ }\frac{dx}{\sqrt{36-{x}^{2}}}\ =\ \frac{1}{6}\int\limits_{\ }^{\ }\frac{dx}{\sqrt{1\ -\ {\left({\frac{x}{6}}\right)}^{2}}}\$

Going ahead with the method of integration by substitution, put $\ \frac{x}{6}\ =\ t$ Differentiating $\ \frac{x}{6}\ =\ t$ , we get

$dx\ =\ 6dt$

Changing the variable in the integral, we get

I1 = $\frac{1}{6}\int\limits_{\ }^{\ }\frac{dx}{\sqrt{1\ -\ {\left({\frac{x}{6}}\right)}^{2}}}\ =\ \frac{1}{6}\int\limits_{\ }^{\ }\frac{6dt}{\sqrt{1\ -\ {\left({t}\right)}^{2}}}$

$=\int\limits_{\ }^{\ }\frac{dt}{\sqrt{1\ -\ {\left({t}\right)}^{2}}}$

The integral we have now, we use the formula $\int\limits_{\ }^{\ }\frac{dt}{\sqrt{1\ -\ {\left({t}\right)}^{2}}}\ =$ $arcsin\ t\ +c\$

Putting in the value for t , we obtain the integral

I1 = $arcsin\ \left({\frac{x}{6}}\right)\ +{C}_{1}\$

Step 3 of 3

Chapter 7.3, Problem 12E is Solved

Textbook: Calculus: Early Transcendentals

Edition: 1

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

ISBN: 9780321570567

Since the solution to 12E from 7.3 chapter was answered, more than 256 students have viewed the full step-by-step answer. This full solution covers the following key subjects: evaluate, Integrals. This expansive textbook survival guide covers 85 chapters, and 5218 solutions. The answer to “Evaluate the following integrals.” is broken down into a number of easy to follow steps, and 4 words. The full step-by-step solution to problem: 12E from chapter: 7.3 was answered by Sieva Kozinsky, our top Calculus solution expert on 03/03/17, 03:45PM. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. Calculus: Early Transcendentals was written by Sieva Kozinsky and is associated to the ISBN: 9780321570567.