The University of Akron
Department of Mathematics
Calculus III
Spring 2026, Golovaty

Review Problems for Test 3 - Part II

Problem 1. Find the volume of the oblique segment of a paraboloid bounded by the paraboloid $z=x^2+y^2$ and the plane $z=y+2$.

Problem 2. Compute by triple integration the volume of the region $T$ that is bounded by the parabolic cylinder $x=y^2$ and the planes $z=0$ and $x+z=1$.

Problem 3. Compute the value of the triple integral $$\int\int\int_Tx^2\,dV,$$ where T is the tetrahedron bounded by the coordinate planes and the first-octant part of the plane with the equation $x+y+z=1$.

Problem 4. Compute the value of the triple integral $$\int\int\int_T(x+y)\,dV,$$ where T is the region between the surfaces $z=2-x^2$ and $z=x^2$ for $0\leq y\leq 3$.

Problem 5. Find the mass of the solid bounded by the surfaces $z=x^2$, $y+z=4$, $y=0$, and $x=0$ if the density of the solid is given by $\rho(x,y,z)=x$.

Problem 6. Use cylindrical coordinates to find the volume of the region bounded by the parabolids $z=2x^2+y^2$ and $z=12-x^2-2y^2$.

Problem 7. Use cylindrical coordinates to find the triple intergal $$\int\int\int_Tz\,dV,$$ where $T$ is bounded by the plane $z=0$ and the paraboloid $z=9-x^2-y^2$.

Problem 8. Describe the surface $\rho=2a\sin{\phi}$ and compute the volume of the region it bounds.

Problem 9. Describe the surface $\rho=1+\cos{\phi}$ and compute the volume of the region it bounds.

Problem 10. Solve $u=x-2y,\ v=3x+y$ for $x$ and $y$ in terms of $u$ and $v$. Then compute the Jacobian $\partial(x,y)/\partial(u,v)$.

Problem 11. Let $R$ be the parallelogram bounded by the lines $x+y=1$, $x+y=2$, $2x-3y=2$, and $2x-3y=3$. Substitute $u=x+y,\ v=2x-3y$ to find its area $$A=\int\int_A\,dA$$.

Problem 12. Substitute $u=xy$ and $v=y/x$ to find the area of the first quadrant region bounded by the lines $y=x$, $y=2x$, and the hyperbolas $xy=1,\ xy=2$.