91Ó°ÊÓ

Use double integrals to find the area of the given region. $$ \text { The region inside the cardioid } r=a(1+\cos \theta) \text { and outside the circle } r=a \text {. } $$

Find the area of the surface in the first octant which is cut from the cone \(x^{2}+y^{2}=z^{2}\) by the plane \(x+y=4\).

Use double integrals to find the area of the given region. $$ \text { The region inside the circle } r=1 \text { and outside the lemniscate } r^{2}=\cos 2 \theta \text {. } $$

Find the mass and center of mass of the given lamina if the area density is as indicated. Mass is measured in slugs and distance is measured in feet. A lamina in the shape of the region bounded by the curve \(y=e^{x}\), the line \(x=1\), and the coordinate axes. The area density varies as the distance from the \(x\) axis.

$$ \int_{-1}^{1} \int_{1}^{e x} \frac{1}{x y} d y d x $$

$$ \int_{0}^{1} \int_{y^{2}}^{y} \sqrt{\frac{y}{x}} d x d y $$

Find the volume of the solid enclosed by the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) by using (a) cylindrical coordinates and (b) spherical coordinates.

Evaluate the triple integral. \(\iiint_{S} y d V\) if \(S\) is the region bounded by the tetrahedron formed by the plane \(12 x+20 y+15 z=60\) and the coordinate planes.

$$ \int_{0}^{1} \int_{x^{2}}^{x} \sqrt{\frac{y}{x}} d y d x $$

Find the area of the portion of the surface of the sphere \(x^{2}+y^{2}+z^{2}=36\) which lies within the cylinder \(x^{2}+y^{2}=9\).