Unit 4 test: Multiple Integrals

Evaluate $\displaystyle\iint_R (x + 2y)\,dA$, where $R = [0, 2] \times [0, 1]$.

Evaluate $\displaystyle\int_0^1 \int_0^2 xy^2\,dy\,dx$.

Evaluate $\displaystyle\iint_D xy\,dA$, where $D$ is the region between $y = x^2$ and $y = x$ for $0 \le x \le 1$.

Which integral equals $\displaystyle\int_0^1 \int_x^1 f(x, y)\,dy\,dx$ with the order of integration reversed?

Evaluate $\displaystyle\int_0^1 \int_y^1 e^{x^2}\,dx\,dy$ by reversing the order of integration.

Evaluate $\displaystyle\iint_D \left(x^2 + y^2\right)dA$, where $D$ is the disk $x^2 + y^2 \le 9$.

Evaluate $\displaystyle\iint_D xy\,dA$, where $D$ is the part of the disk $x^2 + y^2 \le 4$ in the first quadrant.

A lamina occupies the triangle $x \ge 0$, $y \ge 0$, $x + y \le 1$ and has density $\rho(x, y) = x$. Find its mass.

Find the centroid $(\bar{x}, \bar{y})$ of the upper half of the unit disk, $x^2 + y^2 \le 1$, $y \ge 0$.

Evaluate $\displaystyle\iiint_B xyz\,dV$, where $B = [0, 1] \times [0, 2] \times [0, 3]$.

Find the volume of the tetrahedron bounded by the coordinate planes and the plane $x + 2y + 3z = 6$.

Find the volume of the solid between the paraboloids $z = x^2 + y^2$ and $z = 2 - x^2 - y^2$.

What is the volume element $dV$ in spherical coordinates $(\rho, \theta, \phi)$?

Evaluate $\displaystyle\iiint_B \sqrt{x^2 + y^2 + z^2}\,dV$, where $B$ is the unit ball.

Let $R$ be the rectangle bounded by $x + y = 0$, $x + y = 2$, $x - y = 0$ and $x - y = 4$. Use $u = x + y$, $v = x - y$ to evaluate $\displaystyle\iint_R (x - y)\,dA$.