Problem 18 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \( \left. (x + \frac{2}{3}x^3) \right |_{y}^{2y} = 2y + \frac{16}{3}y^3 - (y + \frac{2}{3}y^3) = y + \frac{14}{3}y^3\).

Step 2: 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.

Step 3: 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: \(\frac{1}{2} + \frac{14}{12} + \frac{2}{3} = \frac{25}{6}\).