Question

The mass of the solid of revolution obtained by rotating the graph of y = 4.5 − 0.5$x^2$ on [0, 3] around the y-axis and whose density at height y is ρ( y) = 1.3 − 0.233y ounces per cubic inch.

Short answer

$6\pi$

Step-by-step solution

Step1. Given Information

Equation of graph,

$y=4.5-0.5x^2$

Equation of density,

$\rho \left(y\right)=1.3-0.233y$

Step2. Calculate

$$\begin{gathered} M={\int }_0^{2\pi }{\int }_0^{\pi }{\int }_1^2\rho \sin \varphi d\rho d\varphi d\theta \\ \iint \frac{3}{2}\sin \varphi d\varphi d\theta =3\int d\theta =6\pi \end{gathered}$$