Exam 13: Double and Triple Integrals

Find the volume of the solid bounded by the given function over the specified region $\Omega$ . $f ( x , y ) = 3 x - y + 5$ and $\Omega = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 0 \leq \mathrm { y } \leq 2 x \}$

Find the volume of the solid bounded above by the plane $z = x + y$ and below by the triangle with vertices (0, 0, 0), (1, 1, 0), and (1, -1, 0).

Let T be the triangle with vertices (0, 0), (2, 4), and (2, 0). Let the density at each point of T be equal to 1. Find $I _ { y }$ for T.

Find the area between the cardioids $r = 3 + 3 \sin ( \theta )$ and $r = 2 + 2 \sin ( \theta )$