Question

The value of the integral \[ \int_{-1}^{1} \int_{-1}^{1} |x + y| \, dx \, dy \] (round off to 2 decimal places) is ................

Hint:

When integrating functions with absolute values, break the region into intervals where the integrand's expression inside the absolute value changes sign.

Answer 1

Correct Answer: 2.6

Step 1: Break the absolute value into cases.
We split the integral into regions where the expression inside the absolute value changes sign. The function \( |x + y| \) can be written as: \[ |x + y| = \begin{cases} x + y, & \text{if } x + y \geq 0,
-(x + y), & \text{if } x + y<0. \end{cases} \] We divide the integral based on these cases, where the boundary line is \( x + y = 0 \).
Step 2: Setup the integrals.
We break the square integral into regions. For each region, the integrand is either \( x + y \) or \( -(x + y) \), and we compute the integral in each region. - For \( x + y \geq 0 \), the integral becomes \( \int_0^1 \int_0^1 (x + y) \, dx \, dy \). - For \( x + y<0 \), the integral becomes \( \int_0^1 \int_0^1 (-(x + y)) \, dx \, dy \).
Step 3: Calculate the integral.
After calculating these integrals and evaluating the limits, we get: \[ \boxed{8.00}. \]