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Chapter 4: Q. 45 (page 373)

Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises 19鈥�64. Use
a graph to check your answer. (Hint: The integrands that involve
absolute values will have to be considered piecewise.)
${鈭�}_{0}^{\frac{3}{2}} \frac{1}{\sqrt{9 - x^{2}}} d x$

Short Answer

Expert verified

${鈭�}_{0}^{\frac{3}{2}} \frac{1}{\sqrt{9 - x^{2}}} d x = \frac{蟺}{6}$ .

Step by step solution

01

Step 1. Given information.

A definite integral is given as ${鈭�}_{0}^{\frac{3}{2}} \frac{1}{\sqrt{9 - x^{2}}} d x$ .

02

Step 2. Using the Fundamental Theory of Calculus.

We get

${鈭�}_{0}^{\frac{3}{2}} \frac{1}{\sqrt{9 - x^{2}}} d x = {鈭�}_{0}^{\frac{3}{2}} \frac{1}{3 \sqrt{1 - {(\frac{x}{3})}^{2}}} d x = \frac{1}{3} {鈭�}_{0}^{\frac{3}{2}} \frac{1}{\sqrt{1 - {(\frac{x}{3})}^{2}}} d x = \frac{1}{3} (3) {[\sin^{- 1} \frac{x}{3}]}_{0}^{\frac{3}{2}} = \sin^{- 1} (\frac{\frac{3}{2}}{3}) - \sin^{- 1} 0 = \sin^{- 1} (\frac{1}{2}) - 0 = \frac{蟺}{6} - 0 = \frac{蟺}{6}$

So the exact value of the given definite integral is $\frac{蟺}{6}$ .

03

Step 3. The graph to verify the answer is

The solution is the area under graph which is

a 鈮� 0.5235987 鈮� \frac{蟺}{6}

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Most popular questions from this chapter

Find the sum or quantity without completely expanding or calculating any sums.

Given ${鈭�}_{k = 0}^{25} k = 325$ and ${鈭�}_{k = 3}^{28} {(k - 3)}^{2} = 14, 910$ , find the value of ${鈭�}_{k = 3}^{25} (k^{2} - 5 k + 9)$ .

Explain why we call the collection of antiderivatives of a function f a family. How are the antiderivatives of a function related?

Describe the intervals on which the function f is positive, negative, increasing and decreasing. Them describe the intervals on which the function A is positive , negative, increasing and decreasing

Show by exhibiting a counterexample that, in general, $鈭� f (x) g (x) d x 鈮� (鈭� f (x) d x) (鈭� g (x) d x)$ . In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

Find the sum or quantity without completely expanding or calculating any sums.

Given ${鈭�}_{k = 3}^{10} a_{k} = 12$ and ${鈭�}_{k = 2}^{10} a_{k} = 23$ , find $a_{2}$ .

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