Question

Evaluate the partial integral. $$ \int_{0}^{\sqrt{4-x^{2}}} x^{2} y d y $$

Short answer

The result of evaluating the given integral is \(2 x^2 - (1/2) x^4\).

Step-by-step solution

Perform the integration

As x is being treated as a constant here, the integral of \(x^{2} y\) with respect to y will be \((1/2) x^2 y^2\). Verification of this can be made by differentiating \((1/2) x^2 y^2\) with respect to y which will give back the original integrand, \(x^{2} y\).

Substitute the limits of integration

After finding the integral, it's necessary to substitute the upper and lower limits of integration, \(\sqrt{4-x^{2}}\) and 0, respectively, for y. The result of substituting the upper limit is \((1/2) x^2 (\sqrt{4-x^{2}})^2 = (1/2) x^2 (4-x^{2})\). Substituting the lower limit (0), results in 0 no matter the value of x. As a final result, subtract the result of substitifying through the lower limit from the output of substituting the upper limit to get \((1/2) x^2 (4-x^{2})\).

Simplify the expression

Simplify \((1/2) x^2 (4-x^{2})\) which is equal to \(2 x^2 - (1/2) x^4\).