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Chapter 7: Problem 3

What change of variables is suggested by an integral containing \(\sqrt{100-x^{2}} ?\)

Short Answer

Expert verified
Answer: The suggested change of variables is to use the trigonometric substitution \(x = 10\sin{胃}\).

Step by step solution

01

Identify Trigonometric Substitution

Recall the Pythagorean identity: \(\sin^2{胃} + \cos^2{胃} = 1\). The given integral contains the square root of a number minus the square of the variable, which resembles this identity. Thus, it is a good idea to use trigonometric substitution. Let's start with the substitution \(x = 10\sin{胃}\), so we have:\[x^{2} = 100\sin^{2}{胃}\] Now, we need to rewrite the radical in terms of \(\sin{胃}\).
02

Rewrite radical expression using trigonometric identity

We have the expression \(\sqrt{100-x^{2}}\). By using the substitution \(x=10\sin{胃}\) and the Pythagorean identity, we can rewrite the expression as follows: \[\sqrt{100 - x^2} = \sqrt{100 - 100\sin^2{胃}}\] \[\sqrt{100 - x^2} = \sqrt{100(1 - \sin^2{胃})}\] \[\sqrt{100 - x^2} = \sqrt{100\cos^2{胃}}\] \[\sqrt{100 - x^2} = 10|\cos{胃}|\] Now, we have successfully rewritten the radical expression in terms of \(\cos{胃}\).
03

Calculate derivative of substitution

The last thing we need to do is find the derivative of our substitution variable, \(x\), with respect to the new variable, \(胃\). \[\frac{dx}{d胃} = 10\cos{胃}\] Now we have everything we need to make the substitution in our integral.
04

Conclusion: Change of Variables

To simplify the integral containing \(\sqrt{100-x^2}\), the suggested change of variables is to use trigonometric substitution with \(x=10\sin{胃}\). After making this substitution, we get: \[\sqrt{100-x^2}= 10|\cos{胃}|\] \[\frac{dx}{d胃}=10\cos{胃}\] With this change of variables, the integral should be easier to solve.

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