Question

In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{2} \int_{y}^{2 y}\left(10+2 x^{2}+2 y^{2}\right) d x d y $$

Short answer

The value of the iterated integral is \( 20 + \frac{56}{3} = \frac{116}{3} \). So, the answer is \( \frac{116}{3} \).

Step-by-step solution

Integrate with respect to x

First, perform the inner integral with respect to 'x' from 'y' to '2y', treating 'y' as a constant. \[ \int_{y}^{2 y}\left(10+2 x^{2}+2 y^{2}\right) d x = [10x + \frac{2}{3} x^{3} + 2 y^{2}x]_{y}^{2 y} \]

Substitute the limits of x

Next, substitute the limits of x into the integrated function: \[ = [20y + \frac{16}{3} y^{3} + 4 y^{3}] - [10y + \frac{2}{3} y^{3} + 2 y^{3}] = 10y+ \frac{14}{3}y^{3} \]

Integrate with respect to y

Now, perform the outer integral operation with respect to 'y' from '0' to '2': \[ \int_{0}^{2} (10y + \frac{14}{3}y^{3}) d y = [5y^{2} + \frac{14}{12}y^{4}]_{0}^{2} \]

Substitute the limits of y

Finally, substitute the limits of y into the integrated function: \[ = [5(2)^{2} + \frac{14}{12}(2)^{4}] - 0 = 20 + \frac{56}{3} \]