91Ó°ÊÓ

Find the area of the surface which is cut from the plane \(2 x+y+z=4\) by the planes \(x=0, x=1, y=0\), and \(y=1\).

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 in the first quadrant bounded by the parabola \(y=x^{2}\), the line \(y=1\), and the \(y\) axis. The area density at any point is \((x+y)\) slugs/ft \(^{2}\).

Evaluate the iterated integral. $$ \int_{0}^{\pi / 4} \int_{2 \sin \theta}^{2 \cos \theta} \int_{0}^{r \sin \theta} r^{2} \cos \theta d z d r d \theta $$

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 \(x+y=4\).

The line segment from the origin to the point \((a, b)\) is revolved about the \(x\) axis. Find the area of the surface of the cone generated.

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 in the first quadrant bounded by the circle \(x^{2}+y^{2}=4\) and the line \(x+y=2\). The area density at any point is \(x y\) slugs/ft \(^{2}\).

Derive the formula for the area of the surface of a sphere by revolving a semicircle about its diameter.

Approximate the volume of the solid bounded by the surface \(100 z=300-25 x^{2}-4 y^{2}\), the planes \(x=-1, x=3, y=-3\), and \(y=5\), and the \(x y\) plane. To find an approximate value of the double integral take a partition of the region in the \(x y\) plane by drawing the lines \(x=1, y=-1, y=1\), and \(y=3\), and take \(\left(\xi_{i}, \gamma_{i}\right)\) at the center of the \(i\) th subregion.

The loop of the curve \(18 y^{2}=x(6-x)^{2}\) is revolved about the \(x\) axis. Find the area of the surface of revolution generated.

Use triple integration. $$ \text { Find the volume of the solid enclosed by the sphere } x^{2}+y^{2}+z^{2}=a^{2} \text {. } $$