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((15)/(17)-\frac13\right)=\boxed{(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{(28\pi)/(51)}`