Task 29.1

This task has you practice setting up triple integrals.

  1. The iterated triple integral $\ds\int_{-1}^1\int_0^4\int_0^{y^2}dzdxdy$ gives the volume of the solid $D$ that lies under the surface $z=y^2$, above the $xy$-plane, and bounded by the planes $y=-1$, $y=1$, $x=0$, and $x=4$. Sketch this region.
  2. Set up an iterated triple integral that gives the volume of the solid in the first octant that is bounded by the coordinate planes ($x=0$, $y=0$, $z=0$), the plane $y+z=2$, and the surface $x=4-y^2$, using the order of integration $dxdzdy$. Make sure you sketch the region.
  3. Set up an integral to give the volume of the pyramid in the first octant that is below the planes $\ds\frac{x}{3}+\frac{z}{2}=1$ and $\ds\frac{y}{5}+\frac{z}{2}=1$. [Hint, don't let $z$ be the inside bound. Try an order such as $dydxdz$.]
  4. (Optional Challenge) Set up an iterated triple integral that gives the volume of the region $D$ that is inside both right circular cylinders $x^2+z^2=1$ and $y^2+z^2=1$. Don't forget to draw the region.

Task 29.2

We've now found the mass and center-of-mass for straight wires, thin flat metal plates, and solid regions in space. Earlier in the semester we used $$s = \int_C ds = \int_C \left|\frac{d\vec r}{dt}\right|dt$$ to obtain the length of a thin wire lying on the curve $C$ with parametrization $\vec r(t)$. For such a wire, we use the differential $$\underbrace{ds}_{\text{little distance}} = \underbrace{ \left| \frac{d\vec r}{dt}\right| }_{ \text{speed} }\underbrace{dt}_{\text{little time}}$$ instead of $dx$ (little length in a straight rod), $dA$ (little area in a thin metal plate), or $dV$ (little volume in a solid). The differential $ds$ can replace $dx$, $dA$, or $dV$ in any of our previous formulas to help us determine, for a curved wire, the length, mass, center-of-mass, and more. This task has you set up several integrals that do this.

Consider a wire that lies along the curve $C$ with parametrization $\vec r(t) = (5\cos t,5\sin t)$ for $0\leq t\leq\pi$.

  1. Draw the curve, compute $\frac{d\vec r}{dt}$, and show that $ds = 5 dt$.
  2. Evaluate $\int_C ds$ to obtain the length of the wire. Since the curve is half a circle, the length you obtain from integration should be half the circumference of the circle.
  3. Assuming the density is constant, why do we know $\bar x=0$?
  4. Set up an integral formula for $\bar y$ and compute the integrals involved to obtain $\bar y$, showing your integration steps.
  5. If instead, the density is $\delta = xy^2+7$, then set up an integral formula to find $\bar x$. Use software to compute the integral.

Task 29.3

A sphere of radius $a$ centered at the origin is described by the equation $x^2+y^2+z^2 = a^2$. A right circular cone whose tip is at the origin is given by $z=\sqrt{x^2+y^2}$ or $z^2=x^2+y^2$.

  1. Draw the surface $x^2+y^2+z^2 = a^2$ and then set up an iterated triple integral using Cartesian coordinates to compute the volume inside the sphere $x^2+y^2+z^2=a^2$.
  2. Draw the surface $z^2=x^2+y^2$ and then set up an iterated triple integral using Cartesian coordinates to compute the volume of the solid cone that lies above the cone $z^2=x^2+y^2$ and below the plane $z=h$.
  3. Use software to compute both integrals above. If software can't compute one of these integrals (the program hangs, never finishes, etc.), it's OK. These integrals are brutal, and we'll soon learn that different coordinate systems will make quick work of these integrals.

Task 29.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.