Section 18: Double integrals over regions
Problem types
- Volume
- Advanced Double Integrals
- Geometry/Symmetry
Volume
Find the volume of the given solid.
- Under the plane $x-2y+z =1$ and above the region in the $xy$-plane bounded by $x+y
=1$ and $x^2 +y =1$.
- Bounded by the cylinders $x^2+y^2 = r^2$ and $y^2 + z^2 = r^2$.
Advanced Double Integrals
Evaluate the double integrals:
- $\int_0^3 \int_{\sqrt{x/3}}^1 e^{y^3} dy dx$
- $\int_0^1 \int_0^y e^{-y^2} dx dy$
- $\int_0^4 \int_y^4 \frac{y^3}{x} e^{x^2} dx dy$
- $\int_0^{\sqrt{\pi}} \int_x^{\sqrt{\pi}} \cos(y^2) dy dx$
Geometry/Symmetry
Use geometry, symmetry, or both, to evaluate the double integral.
- $\iint_D (x+2) dA$, $D = \{(x,y): 0 \leq y \leq \sqrt{9-x^2}\}$
- $\iint _D \sqrt{R^2 -x^2 -y^2} dA$, where $D$ is the disk with center the origin and radius $R$.