1. In Exercises 8 - 11, a region of the Cartesian plane is shaded. Use the Disk/Washer Method to find the volume of the solid of revolution formed by revolving the region about the y-axis.

Manual Solution

\[ y = 5x \] Make it a function of \(y\):

\[ x = \frac{y}{5}\] The disk method is given as:

\[ V = \pi \int_a^b{R(x)^2}dx\]

\[ \begin{align} V &= \pi \int_5^{10}{(\frac{y}{5})^2}dx \\ \\ &= \pi\bigg[ \frac{x^3}{75} \bigg]_5^{10} \\ \\ &= \pi \bigg[\frac{1000}{75} - \frac{125}{75}\bigg] \\ \\ &= \pi \frac{875}{75} \\ \\ &\approx 36.65191 \space units^3 \end{align} \]

Solution in R

volume <-  integrate( function(y){(y/5)^2} ,lower = 5, upper = 10)

print(paste0("The volume is ", volume$value * pi , " cubic units."))
## [1] "The volume is 36.6519142918809 cubic units."