21. Multiple Integrals in Curvilinear Coordinates

Homework

1. Find the mass of a plate inside the circle \(r=3\cos\theta\) if the density is \(\delta=r\).

   

2. Find the volume under the function \(\displaystyle f=x^2+y^2\) over the region bounded by \(y=x\), \(y=0\) and \(y=\sqrt{16-x^2}\)

3. Find the area inside the inner loop of the limacon \(r=1-2\cos\theta\).

   

4. Consider the plate inside the cardioid \(r=1-\cos\theta\), with a surface density of \(\delta(x,y)=|y|\).

   

   1. Find the total area of the plate.

   2. Find the centroid of the plate. Use symmetry to find one of \(\bar x_\text{cent}\) or \(\bar y_\text{cent}\).

   3. Find the total mass of the plate if its surface density is \(\delta(x,y)=|y|\).
      Hint: Find the mass of the upper half and double it.

   4. Find the center of mass of the plate.
      Hint: Use symmetry to find \(M_y\). Again find \(M_x\) for the upper half and double it.

5. Compute the integral \(\displaystyle I=\iiint z^2\,dV\) over the cone with radius \(4\) and height \(4\).

   

6. Consider the solid under the paraboloid \(z=16-x^2-y^2\), with a volume density \(\delta(x,y,z)=x^2+y^2\).

   

   1. Find the volume of the paraboloid.

   2. Find the centroid of the paraboloid. Use symmetry to find \(\bar x_\text{cent}\) and \(\bar y_\text{cent}\).

   3. Find the mass of the paraboloid.

   4. Find the center of mass of the paraboloid.

7. Find the volume of the jello mold whose upper surface is given in cylindrical coordinates by: \[ z=4r-r^2 \]

   

8. An igloo is an ice shell filling the region between \(2\) concentric hemispheres with radii \(\rho_1=10\,\text{ft}\) and \(\rho_2=12\,\text{ft}\).

   

   1. Find the volume of the igloo.

   2. Find the centroid of the igloo. Use symmetry to find \(\bar x_\text{cent}\) and \(\bar y_\text{cent}\).

   3. If the temperature of the ice in the igloo is \(T=\left(32-\dfrac{4}{\rho}\right){}^\circ\text{F}\), find the average temperature of the ice.

9. An apple is made by rotating the spiral \(\rho=\phi\) about the \(z\)-axis. The density is \(\delta=\rho\). Give both exact and decimal answers, so you can see how the center of mass compares to the centroid.

   

   1. Find the volume of the apple.
      (You must do the integral.)

   2. Find the centroid of the apple. Use symmetry to find \(\bar x_\text{cent}\) and \(\bar y_\text{cent}\).
      (Reduce the triple integral to a single integral. Then, you may use a computer to do the integral.)

   3. Find the mass of the apple.
      (Reduce the triple integral to a single integral. Then, you may use a computer to do the integral.)

   4. Find the center of mass of the apple. Use symmetry to find \(\bar x_\text{cm}\) and \(\bar y_\text{cm}\).
      (Reduce the triple integral to a single integral. Then, you may use a computer to do the integral.)

10. Consider the coordinate system where \(u\) and \(v\) are defined by: \[\begin{aligned} u&=x^3+y^3 \qquad (1) \\ v&=2x^3+y^3 \qquad (2) \end{aligned}\] Find the Jacobians \(\dfrac{\partial(x,y)}{\partial(u,v)}\) and \(\dfrac{\partial(u,v)}{\partial(x,y)}\) and show they are reciprocals.
    HINT: First find \(x\) and \(y\) as functions of \(u\) and \(v\).

11. Compute \(\displaystyle I=\iint_R x\,dx\,dy\) where \(R\) is the region bounded by \[\begin{aligned} y&=2x+x^2 \qquad &y&=4x-x^2 \\ y&=4x+x^2 \qquad &y&=8x-x^2 \\ \end{aligned}\] HINTS: Define \(u\) and \(v\) so that \[ y=ux+x^2 \qquad y=vx-x^2 \] What are the values of \(u\) and \(v\) on the \(4\) boundaries?

    

12. Compute the mass inside the ellipse \(4x^2+16y^2=36\) if the density is \(\delta=4x^2+16y^2\).

13. Compute the Jacobian factor for the following 3D coordinate systems: \[ x=(u+v)\cos\theta \qquad y=(u+v)\sin\theta \qquad z=u-v \]