Question

etermine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(x=\sin \theta,\) then \(\int_{-1}^{1} x^{2} \sqrt{1-x^{2}} d x=2 \int_{0}^{\pi / 2} \sin ^{2} \theta \cos ^{2} \theta d \theta\)

Short answer

The statement is false, because after substituting the function and simplifying, the equation obtained does not match the original equation in the problem statement. The reason is that the term \(\cos^2 (\theta)\) from the original statement is not present in the simplified equation, which has \(\cos (\theta)\) instead.

Step-by-step solution

Variable Substitution

Substitute \(x = \sin \theta\) into the integral \(\int_{-1}^{1} x^{2} \sqrt{1-x^{2}} dx\), to obtain: \(\int_{-\pi/2}^{\pi/2} \sin^2 \theta \cdot \cos \theta d\theta\). Note that the limits have changed because the values of \(x = \sin \theta\) range from \(-1\) to \(1\) when \(\theta\) varies from \(-\pi/2\) to \(\pi/2\).

Simplify the Equation

Simplify the equation, which is the integral from \(-\pi/2\) to \(\pi/2\) of \(\sin^2 \theta \cdot \cos \theta d \theta\). Here we double the integral from 0 to \(\pi/2\) because the function \(\sin^2 \theta \cdot \cos \theta\) is symmetric around the y-axis, resulting in: \(2\int_{0}^{\pi / 2} \sin ^{2} \theta \cos \theta d \theta\).

Compare both Sides

Now, it can be noticed that the simplified equation obtained from the substitution is not identical to the equation given in the problem statement, \((2\int_{0}^{\pi / 2} \sin ^{2} \theta \cos ^{2} \theta d \theta)\). The original equation has \(\cos^2 \theta\), whereas the equation obtained through substitution has \(\cos \theta\).