Question

Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and lines about the​ y-axis. =-x^2 , y=6 - 5 x​, x=​0, for x>=0

Answer 1

The two curves intersect when

$-x^2=6-5x\implies x^2-5x+6=(x-3)(x-2)=0\implies x=2,x=3$

The region bounded by $x=0$ and the two curves "ends" with a vertex where $x=2$. So over the interval [0, 2], we have $6-5x\ge-x^2$, so that the volume is

$\displaystyle2\pi\int_0^2x(6-5x+x^2)\,\mathrm dx=\boxed{\frac{16\pi}3}$

That is, each cylindrical shell has a radius of $x$ and height $(6-5x)-(-x^2)=6-5x+x^2$, and their contribution to the total volume is $2\pi$ times the radius and height.