Question

When the shaded region below is rotated about the x-axis the resulting solid is a circular cone with radius 2 m and height 3 m. Use the disk or shell method to find the volume of this cone

Answer 1

Answer:

Therefore, the volume of the cone is V=4π.

Step-by-step explanation:

From task we have a circular cone with radius 2 m and height 3 m. We use the disk method to find the volume of this cone.  

We have the formula:

$\boxed{V=\int_0^h\pi \cdot \left(\frac{r}{h}x\right)^2\, dx}$

We know that r=2 and h=3, and we get:

$V=\int_0^3\pi \cdot \left(\frac{2}{3}x\right)^2\, dx\\\\V=\int_0^3 \pi \frac{4}{9}x^2\, dx\\\\V= \frac{4\pi}{9} \int_0^3 x^2\, dx\\\\V= \frac{4\pi}{9} \left[\frac{x^3}{3}\right]_0^3\, dx\\\\V= \frac{4\pi}{9}\cdot 9\\\\V=4\pi$

Therefore, the volume of the cone is V=4π.