Question

Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If​possible, find the volume of S by both the​disk/washer and shell methods. Check that your results agree.y=​x,y= x^1/15in the first​ quadrant; revolved about the​ x-axisFind the volume using the​ disk/washer method.Volume =

Answer 1

The two curves $y=x$ and $y=x^{1/15}$ intersect at $x=0$ and $x=1$, with $x^{1/15}\ge x$ over $0\le x\le1$.

• Washer method

$\displaystyle\pi\int_0^1\left(\left(x^{1/15}\right)^2-x^2\right)\,\mathrm dx=\pi\int_0^1\left(x^{2/15}-x^2\right)\,\mathrm dx=\pi\left(\frac{15}{17}-\frac13\right)=\boxed{\frac{28\pi}{51}}$

• Shell method

We have $y=x^{1/15}\implies x=y^{15}$. The curves $x=y$ and $x=y^{15}$ intersect at $y=0$ and $y=1$, with $y\ge y^{15}$ over $0\le y\le1$.

$\displaystyle2\pi\int_0^1y\left(y-y^{15}\right)\,\mathrm dy=2\pi\int_0^1\left(y^2-y^{16}\right)\,\mathrm dy=2\pi\left(\frac13-\frac1{17}\right)=\boxed{\frac{28\pi}{51}}$