Use the Theorem of Pappus to find the volume of the solid formed by revolving the region bounded by the graphs of $y = x , y = 10$, and $x = 0$ about the $x$-axis. Round your answer to two decimal places.
A) $1809.56$
B) $2787.64$
C) $2094.40$
D) $3141.59$
E) $3619.11$

Question

Quizplus · Accepted Answer

The Theorem of Pappus states that the volume of a solid of revolution generated by revolving a plane figure about an external axis is equal to the product of the area of the figure and the distance traveled by its centroid. The region described is a right triangle with base 10 (along the y-axis from y=0 to y=10) and height 10 (along the x-axis from x=0 to x=10). The area of this triangle is $A = \frac{1}{2} \times base \times height = \frac{1}{2} \times 10 \times 10 = 50$. The centroid of a right triangle is located at one-third the base and one-third the height from the right angle, so the centroid is $\frac{10}{3}$ units from the x-axis. The distance traveled by the centroid when the region is revolved around the x-axis is $2\pi \times \frac{10}{3}$. Therefore, the volume is $V = A \times 2\pi \times \frac{10}{3} = 50 \times 2\pi \times \frac{10}{3} = \frac{1000\pi}{3} \approx 2094.40$.

Use the Theorem of Pappus to Find the Volume of the Solid