Question

a rectangle with vertices A(6,0), K(0,0), L(0,9), and M(6,9) is rotated around the x-axis.To the nearest tenth of a cubic unit, what is the volume of the resulting three-dimensional figure? Approximate pi as 3.14

A.339.12 units^3
B.508.7 units^3
C.1017.4 units^3
D.1526.0 units^3

Answer 1

Look at the picture.

The formula of the volume of the solid created by the rotation around to the
x - axis

$y=f(x);\ x\in\left\ \textless \ a;\ b\right\ \textgreater \ \\\\|V|=\pi\int\limits_a^bf^2(x)dx$

We have:

$f(x)=9;\ x\in\left< 0;\ 6\right>$

$\int f^2(x)dx=\int9^2dx=\int81dx=81x\\\\|V|=\pi\int\limits_0^69^2dx=\pi\left81x\right]^6_0=\pi(81\cdot6-81\cdot0)=\pi\cdot486=486\pi\\\\|V|=486\pi\approx486\cdot3.14=1526.0\ units^3$

Answer: D. 1526.0 units^3.