Question

Let $f(x, y) = x^2 + y^2$ and $R = [-1, 1] \times [0, 1]$, then $\iint_R f(x, y) \, dx \, dy$ is

• A. $\frac{1}{3}$
• B. $\frac{4}{3}$
• C. $\frac{2}{3}$
• D. 1

Hint:

Symmetry trick: Since $x^2$ is an even function of $x$ on $[-1, 1]$, $\int_{-1}^1 (x^2 + y^2) dx = 2 \int_0^1 (x^2 + y^2) dx = 2 [\frac{1}{3} + y^2]$. Then $\int_0^1 2 (\frac{1}{3} + y^2) dy = 2 (\frac{1}{3} + \frac{1}{3}) = \frac{4}{3}$.

Answer 1

The Correct Option is B