Question

An xyz-coordinate system has an x-axis, a y-axis, and a vertical positive z-axis. A right circular cone labeled z = 26 - √( x² + y²) has a circular base in the xy-plane that is a circle centered at (0, 0, 0) and a vertex on the positive z-axis.
Find the volume of the solid bounded by the surface z = ​f(x,y) and the​ xy-plane.
(Type an exact​ answer, using pi as​ needed.)

Answer 1

Answer:

V = 5858.66π

Step-by-step explanation:

This problem can be solved by using the washer method in the integration.

If you assume that the lateral of the cone is given by a line equation of the form:

$z=\frac{r}{h}u$

r: radius of the cone = 26 (because for z=0 -> √( x² + y²)  = 26 = r)

h: height of the cone = 26 (because for x=0 and y=0, z = 26)

you can integrate in the following form to get the volume of the cone:

$V=\pi\int_0^{26}[\frac{r}{h}u]^2du=\pi\frac{r^2}{h^2}[\frac{u^3}{3}]\\\\V=\pi\frac{(26)^2}{(26)^2}\frac{(26)^3}{3}=5858.66\pi$

hence, the volume of the cone is 5858.66π