Question

Evaluate the double integral. $$ \int_{0}^{2} \int_{0}^{\sqrt{1-y^{2}}}-5 x y d x d y $$

Short answer

The value of the double integral is \( 35/3 \) or approximately 11.67.

Step-by-step solution

Solve the inner integral

The first integral to solve is \( \int_{0}^{\sqrt{1-y^{2}}} -5xy dx \). Treating \( y \) as a constant, you would integrate the function with respect to \( x \), yielding a function of \( x \)and \( y \): \( -5x^{2}y/2 = -2.5 x^{2} y \). At the limits from \( 0 \) to \( \sqrt{1-y^{2}} \), the integral results in \(-2.5 (\sqrt{1-y^{2}})^{2} y = -2.5 (1 - y^{2}) y \).

Solve the outer integral

Now, integrate the resulted function in Step 1 with respect to \( y \) over the limit from 0 to 2, thus \( \int_{0}^{2} -2.5 (1 - y^{2}) y dy \). The answer to this integral is \( -2.5y + 5y^{3}/3 \) evaluated from 0 to 2, which gives \( -5 + 80/3 = 35/3 \).