Questions: Use a change of variables to evaluate the following definite integral. ∫[0 to 1] x sqrt(1-x^2) dx Determine a change of variables from x to u. Choose the correct answer below. A. u=x^2 B. u=1-x^2 C. u=sqrt(1-x^2) D. u=2x^1

Solution Steps

To evaluate the integral \(\int_{0}^{1} x \sqrt{1-x^{2}} \, dx\), we need to choose a substitution that simplifies the integrand. The expression under the square root, \(1 - x^2\), suggests that a substitution involving \(u = 1 - x^2\) would be effective.

Let \(u = 1 - x^2\). Then, the derivative of \(u\) with respect to \(x\) is: \[ \frac{du}{dx} = -2x \implies du = -2x \, dx. \] Solving for \(x \, dx\), we get: \[ x \, dx = -\frac{1}{2} du. \]

When \(x = 0\), \(u = 1 - 0^2 = 1\).
When \(x = 1\), \(u = 1 - 1^2 = 0\).
Thus, the limits of integration change from \(x = 0\) to \(x = 1\) to \(u = 1\) to \(u = 0\).

Substituting \(u = 1 - x^2\) and \(x \, dx = -\frac{1}{2} du\) into the integral, we get: \[ \int_{0}^{1} x \sqrt{1-x^{2}} \, dx = \int_{1}^{0} \sqrt{u} \left(-\frac{1}{2}\right) du. \] This simplifies to: \[ -\frac{1}{2} \int_{1}^{0} \sqrt{u} \, du. \]

The integral \(\int \sqrt{u} \, du\) is: \[ \int \sqrt{u} \, du = \int u^{1/2} \, du = \frac{2}{3} u^{3/2} + C. \] Applying the limits of integration: \[ -\frac{1}{2} \left[ \frac{2}{3} u^{3/2} \right]_{1}^{0} = -\frac{1}{2} \left( \frac{2}{3} \cdot 0^{3/2} - \frac{2}{3} \cdot 1^{3/2} \right). \] Simplifying further: \[ -\frac{1}{2} \left( 0 - \frac{2}{3} \right) = -\frac{1}{2} \left( -\frac{2}{3} \right) = \frac{1}{3}. \]

The value of the integral \(\int_{0}^{1} x \sqrt{1-x^{2}} \, dx\) is \(\frac{1}{3}\).

Final Answer