Chapter 5: Q. 13 (page 464)

Explain why, if $x = a \sin u$ , then $\sqrt{a^{2} - x^{2}} = a \cos u$ . Your explanation should include a discussion of domains and absolute values.

Short Answer

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Ans:

$\sqrt{a^{2} - x^{2}} = a \cos u . . . . . [x = a \sin u] = \sqrt{a^{2} - a^{2} \sin^{2} u} = a \cos u$

Step by step solution

Step 1. Given information:

$\sqrt{a^{2} - x^{2}} = a \cos u$

$x = a \sin u$

Step 2. Solving the trigonometric substitution:

If $x = a \sin u$ , then $x$ must be between $- a$ and $a$ . Therefore the substitution $x = a \sin u$ makes sense only for integrands whose domain is contained in [-a, a] .

The domain of $\sqrt{a^{2} - x^{2}}$ is precisely $[- a, a]$ , so the integral involving $\sqrt{a^{2} - x^{2}}$ are well suited for trigonometric substitution with $x = a \sin u$ .

Using the Pythagorean identity $a^{2} \sin^{2} u + a^{2} \cos^{2} u = a^{2}$ ,

$\sqrt{a^{2} - x^{2}} = \sqrt{a^{2} - a^{2} \sin^{2} u} = \sqrt{a^{2} \cos^{2} u} = |a \cos u| [A . \sqrt{A^{2}} = |A|] = a \cos u$

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Explain why, if $x = a \sin u$ , then $\sqrt{a^{2} - x^{2}} = a \cos u$ . Your explanation should include a discussion of domains and absolute values.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving the trigonometric substitution:

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Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving the trigonometric substitution:

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Most popular questions from this chapter

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Explain why, if $x = a \sin u$ , then $\sqrt{a^{2} - x^{2}} = a \cos u$ . Your explanation should include a discussion of domains and absolute values.