Let’s analyze and solve the problems step by step. These problems involve integration in polar coordinates, geometry, Green's theorem, and exact differential forms.

5. (a) Find the area of the region common to the interiors of the cardioids $r = 1 + \cos\theta$ and $r = 1 - \cos\theta$.

• Solution:
  The common region lies where the two cardioids intersect. The cardioids intersect when $1 + \cos\theta = 1 - \cos\theta$, leading to $\cos\theta = 0$. This occurs at $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$.

The area of the common region is given by: $A = \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \left[ (1+\cos\theta)^2 - (1-\cos\theta)^2 \right] \, d\theta$ Simplify: $A = \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \left[ 4\cos\theta \right] \, d\theta$ $A = 2 \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cos\theta \, d\theta$ Evaluate: $\int \cos\theta \, d\theta = \sin\theta$ $A = 2 \left[ \sin\theta \right]_{\frac{\pi}{2}}^{\frac{3\pi}{2}} = 2 \left( \sin\frac{3\pi}{2} - \sin\frac{\pi}{2} \right)$ $A = 2 \left( -1 - 1 \right) = -4$ Since the area cannot be negative, the magnitude is $\boxed{4 \, \text{units}^2}$.

5. (b) Find the volume of the wedge cut from the cylinder $x^2 + y^2 = 1$ by the plane $z = -y$ and $z = 0$.

The cylinder $x^2 + y^2 = 1$ is bounded between $z = -y$ and $z = 0$. Using cylindrical coordinates, $x = r\cos\theta$, $y = r\sin\theta$, $z = z$, and $r = 1$, the limits are:

• $r$ from $0$ to $1$,
• $\theta$ from $0$ to $2\pi$,
• $z$ from $-r\sin\theta$ to $0$.

Volume $V$ is: $V = \int_0^{2\pi} \int_0^1 \int_{-r\sin\theta}^0 r \, dz \, dr \, d\theta$ Integrate with respect to $z$: $\int_{-r\sin\theta}^0 r \, dz = r \left[ z \right]_{-r\sin\theta}^0 = r \left( 0 - (-r\sin\theta) \right) = r^2 \sin\theta$ Integrate with respect to $r$: $\int_0^1 r^2 \sin\theta \, dr = \sin\theta \int_0^1 r^2 \, dr = \sin\theta \left[ \frac{r^3}{3} \right]_0^1 = \frac{\sin\theta}{3}$ Integrate with respect to $\theta$: $\int_0^{2\pi} \frac{\sin\theta}{3} \, d\theta = \frac{1}{3} \int_0^{2\pi} \sin\theta \, d\theta = \frac{1}{3} \left[ -\cos\theta \right]_0^{2\pi} = \frac{1}{3} \left( -\cos(2\pi) + \cos(0) \right) = 0$ Hence, the volume is $\boxed{0}$.

Would you like me to proceed with the remaining parts or clarify any steps? Here's how you can deepen your understanding:

1. Explain how polar integration works for finding areas.
2. Explore why Green's theorem applies to problem 7(a).
3. Discuss the physical interpretation of the cylindrical volume in 5(b).
4. Review the exactness of differential forms for problem 7(b).
5. Dive into spherical coordinates for volume computation in 6(b).

Tip: Break multi-dimensional integrals into small, simple steps and always check the bounds of integration!