91Ó°ÊÓ

Exer. \(11-14:\) Sketch the region of integration for the iterated integral. $$ \int_{-1}^{0} \int_{x}^{-x^{2}} d y d x $$

Exer. \(1-14\) : Describe the graph of the equation in three dimensions. $$ r^{2}+z^{2}=16 $$

Find the area of the surface \(S\). \(S\) is the first-octant part of the hyperbolic paraboloid \(z=x^{2}-y^{2}\) that is inside the cylinder \(x^{2}+y^{2}=1\).

Evaluate the iterated integral. $$ \int_{1}^{2} \int_{-1}^{2}\left(12 x y^{2}-8 x^{3}\right) d y d x $$

Use polar coordinates to evaluate the integral. $$ \begin{aligned} &\iint_{R}\left(x^{2}+y^{2}\right)^{3 / 2} d A\\\ &R \text { is bounded by the circle } x^{2}+y^{2}=4 \text { . } \end{aligned} $$

Sketch the region bounded by the graphs of the equations, and use a triple integral to find its volume. \(y=2-z^{2}, \quad y=z^{2}, \quad x+z=4, \quad x=0\)

A dome-shaped camping tent is designed to have a circular floor of radius 5 feet and a roof described by the graph of \(z=7-\frac{7}{25}\left(x^{2}+y^{2}\right)\) for \(z \geq 0 .\) Approximate the number of square feet of material needed to construct the tent. (Do not consider wasted or overlapping material.)

A homogeneous lamina has the shape of a square of side \(a\). Find the moment of inertia with respect to |al a side (b) a diagonal (c) the center of mass

Evaluate the iterated integral. $$ \int_{0}^{3} \int_{-2}^{-1}\left(4 x y^{3}+y\right) d x d y $$

Sketch the region bounded by the graphs of the equations, and use a triple integral to find its volume. \(z=4 y^{2}, \quad z=2, \quad x=2, \quad x=0\)