Question

what change of variables is suggested by an integral containing $sqrt{100 - x^{2}}$?
choose the correct answer.
a. $x = 10sec\theta$
b. $x = 100sec\theta$
c. $x = 10sin\theta$
d. $x = 10\tan\theta$

Explanation:

Step1: Recall trig - identity

We know the identity $\sin^{2}\theta+\cos^{2}\theta = 1$, or $1-\sin^{2}\theta=\cos^{2}\theta$. If we have $\sqrt{100 - x^{2}}=\sqrt{10^{2}-x^{2}}$, we want to make a substitution such that $10^{2}-x^{2}$ becomes a perfect - square after substitution.

Step2: Make substitution

Let $x = 10\sin\theta$. Then $100 - x^{2}=100-100\sin^{2}\theta=100(1 - \sin^{2}\theta)=100\cos^{2}\theta$. And $\sqrt{100 - x^{2}}=\sqrt{100\cos^{2}\theta}=10|\cos\theta|$.

Answer:

C. $x = 10\sin\theta$