4. In a rectangular coordinate system, lines $p_{1}$ and $p_{2}$ are given by the equations:

$$
p^{\prime}: \quad y=x-4 \quad \text { and } \quad p_{2}: y-2 x+2=0 \text {. }
$$

a) Calculate the area of the figure determined by the lines $p_{1}$ and $p^{\prime}$ and the coordinate axes.

b) Calculate the volume of the solid of revolution that is formed when the triangle bounded by the lines $p_{1}$ and $p^{\prime}$ and the y-axis rotates around the y-axis.
Solution. a) We have

$$
\begin{aligned}
P_{A B C D} & =P_{O A B}-P_{O C D} \\
& =7 \mathrm{~cm}^{2} .
\end{aligned}
$$

The details are left to the reader as an exercise.

b) The desired volume is equal to the difference in volumes of the two cones with vertices
![](https://cdn.mathpix.com/cropped/2024_06_05_e2b4ba6b48251e03981fg-4.jpg?height=492&width=754&top_left_y=1614&top_left_x=726)
$D$ and $B$ (see the diagram on the right). According to this:

$$
V=V_{1}-V_{2}=\frac{2^{2} \pi \cdot 4}{3}-\frac{2^{2} \pi \cdot 2}{3}=\frac{8 \pi}{3}
$$

The details are left to the reader as an exercise.