Question

Evaluate the double integral. $$ \int_{0}^{1} \int_{y}^{2 y}\left(1+2 x^{2}+2 y^{2}\right) d x d y $$

Short answer

The solution to the double integral is \(\frac{25}{6}\).

Step-by-step solution

Integrate with respect to \(x\)

The first step is to perform the inner integral, which means integrating the function with respect to \(x\) first. The antiderivative of the function \(1 + 2x^2\) with respect to \(x\) is \(x + \frac{2}{3}x^3\). The definite integral from y to 2y is: \( \left. (x + \frac{2}{3}x^3) \right |_{y}^{2y} = 2y + \frac{16}{3}y^3 - (y + \frac{2}{3}y^3) = y + \frac{14}{3}y^3\).

Integrate with respect to \(y\)

Next, integrate the resulting function, \(y + \frac{14}{3}y^3 + 2y^2\), with respect to \(y\) from 0 to 1. We will obtain the antiderivative as follows: \(\int_0^1 (y + \frac{14}{3}y^3 + 2y^2)dy = \frac{1}{2}y^2 + \frac{14}{12}y^4 + \frac{2}{3}y^3 |_{0}^{1} = \frac{1}{2} + \frac{14}{12} + \frac{2}{3}\), evaluating the limits yields the final result.

Calculate the final answer

The final results from step 2 are real numbers which need to be added together. The answer to the integral is: \(\frac{1}{2} + \frac{14}{12} + \frac{2}{3} = \frac{25}{6}\).