MA 1024 B '01 Practice Exam 3

  1. Evaluate the following triple integral.

    \begin{displaymath}\int_{-1}^{1}\int_{0}^{1-y^2}\int_{y}^{1-z} 1 \,dx \,dz\,dy \end{displaymath}

  2. Set up, but do not evaluate, a double integral for the volume of the solid that lies under the surface $z=1+x^2+3y^3$ and above the region in the $xy$ plane bounded by the curves $y=0$ and $y=\sin(x)$ for $0
\leq x \leq \pi$. Make sure you include a sketch of the region.

  3. For the following integral, first sketch the domain of integration, then convert the integral to polar coordinates and evaluate the resulting integral.

    \begin{displaymath}\int_{-1}^{1} \int_{0}^{\sqrt{4-x^2}} \sqrt{4-x^2-y^2} \,
dy\, dx \end{displaymath}

  4. Set up, but do not evaluate, a triple integral in cylindrical coordinates that computes the mass of a body bounded by the two surfaces $z= \sqrt{x^2+y^2}$ and $z = 2-x^2-y^2$ if the density is given by $D(x,y,z) = 1+x^2y^2z^2$.

  5. Use a triple integral to find the volume of the solid bounded by $x+3y+4z=12$ and the coordinate planes.

  6. Compute the coordinates of the center of mass, $(\overline{x},\overline{y})$, of a thin plate bounded by the line $y= 3x$ and the parabola $y=4-x^2$ if the density per unit area is $\delta(x,y) =12+ x +y$. Include a sketch of the region.

William W. Farr
2001-12-13