/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 17

Evaluate the following integrals. $$\int \sqrt{64-x^{2}} d x$$

Short Answer

Expert verified
Question: Evaluate the integral \(\int_{0}^{\frac{\pi}{2}} \sqrt{64 - x^2} dx\). Answer: The integral evaluates to 64.

Step by step solution

01

Choose an appropriate trigonometric substitution

In this case, since we have \(\sqrt{64-x^2}\), we can recognize it as a Pythagorean identity if we make the substitution $$x = 8\sin(\theta)$$. This will simplify the square root expression.
02

Differentiate the substitution with respect to the new variable

Now, we will differentiate the chosen substitution to find \(dx\) in terms of \(d\theta\). $$ dx = 8\cos(\theta)d\theta$$
03

Rewrite the integral using the substitution

Plug the substitution into the original integral: $$\int \sqrt{64-x^{2}} dx = \int \sqrt{64-(8\sin(\theta))^2} (8\cos(\theta)d\theta)$$
04

Simplify the integral

Simplify the expression within the integral: $$= \int\sqrt{64 - 64\sin^{2}(\theta)} (8\cos(\theta)d\theta)$$ From the Pythagorean identity, $$\sin^{2}(\theta) + \cos^{2}(\theta) = 1$$, so $$\cos^{2}(\theta) = 1 - \sin^{2}(\theta)$$. Therefore, the expression becomes: $$= \int 8\sqrt{64(1-\sin^{2}(\theta))} (8\cos(\theta)d\theta)$$
05

Evaluate the integral with the simplified expression

Now we have: $$= 64\int_{0}^{\frac{\pi}{2}} \cos(\theta)d\theta$$ Evaluate the integral: $$= 64 [\sin(\theta)]_{0}^{\frac{\pi}{2}}$$ $$= 64 (\sin(\frac{\pi}{2}) - \sin(0))$$ $$= 64(1 - 0)$$ $$= 64$$
06

Convert back to the original variable

Rewrite the final answer in terms of the original variable: $$\int_{0}^{\frac{\pi}{2}} \sqrt{64 - x^2} dx = 64$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t),\) the Laplace transform is a new function \(F(s)\) defined by $$ F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t $$ where we assume that s is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$ F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1} $$ Verify the following Laplace transforms, where a is a real number. $$f(t)=\cos a t \longrightarrow F(s)=\frac{s}{s^{2}+a^{2}}$$

\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t),\) the Laplace transform is a new function \(F(s)\) defined by $$ F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t $$ where we assume that s is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$ F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1} $$ Verify the following Laplace transforms, where a is a real number. $$f(t)=e^{a t} \longrightarrow F(s)=\frac{1}{s-a}$$

By reduction formula 4 in Section 3 $$\int \sec ^{3} u d u=\frac{1}{2}(\sec u \tan u+\ln |\sec u+\tan u|)+C$$ Graph the following functions and find the area under the curve on the given interval. $$f(x)=\left(x^{2}-25\right)^{1 / 2},[5,10]$$

a. Verify the identity \(\sec x=\frac{\cos x}{1-\sin ^{2} x}\) b. Use the identity in part (a) to verify that \(\int \sec x d x=\frac{1}{2} \ln \left|\frac{1+\sin x}{1-\sin x}\right|+C\) (Source: The College Mathematics Joumal \(32,\) No. 5 (November 2001))

When is the volume finite? Let \(R\) be the region bounded by the graph of \(f(x)=x^{-p}\) and the \(x\) -axis, for \(0
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.