Volume of solid rotating y = 5x and y = 5√x about y = 5

[Gundown64]

Homework Statement

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 5x, y = 5$\sqrt{x}$ about y = 5

Homework Equations

A(x)=∏(R²-r²)

The Attempt at a Solution

A(x)=∏(5x)²-(5$\sqrt{x}$)²)
A(x)=∏(25 x² - $\frac{10}{3}$x^{$\frac{3}{2}$})

V=∏$\int$$^{1}_{0}$(25x²- $\frac{10}{3}$x^{$\frac{3}{2}$})dx

V=∏($\frac{25}{3}$x³-$\frac{4}{3}$x^{$\frac{5}{2}$}) {0,1}

V=∏($\frac{25}{3}$-$\frac{4}{3}$)
V= $\frac{21}{3}$∏

Where did I go wrong? I can't figure out the about y=5. If I am correct, you would usually subtract 5 from the two radii, but since they intersect at y=5, the just flip on that intersection point and thus we don't need to find the lost area. Bad explanation, I know, maybe someone can explain it to me.

Thanks in advance!

 

[Mark44]

Homework Statement

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 5x, y = 5$\sqrt{x}$ about y = 5

Homework Equations

A(x)=∏(R²-r²)

The Attempt at a Solution

A(x)=∏(5x)²-(5$\sqrt{x}$)²)

You've made an error right off the bat. I hope you sketched the region being rotated.

You aren't taking into account the fact that the axis of rotation is the line y = 5. The larger radius, R, is 5 - 5x. The smaller radius, r, is 5 - 5√x.

Also, your relevant equation is missing a factor - Δx. If you substitute the values above for R and r, you should get the right result.

A(x)=∏(25 x² - $\frac{10}{3}$x^{$\frac{3}{2}$})

V=∏$\int$$^{1}_{0}$(25x²- $\frac{10}{3}$x^{$\frac{3}{2}$})dx

V=∏($\frac{25}{3}$x³-$\frac{4}{3}$x^{$\frac{5}{2}$}) {0,1}

V=∏($\frac{25}{3}$-$\frac{4}{3}$)
V= $\frac{21}{3}$∏

Where did I go wrong? I can't figure out the about y=5. If I am correct, you would usually subtract 5 from the two radii, but since they intersect at y=5, the just flip on that intersection point and thus we don't need to find the lost area. Bad explanation, I know, maybe someone can explain it to me.

Thanks in advance!

 

[Gundown64]

Ok, that's what my question at the end was about. I wasn't sure if I was supposed to subtract the radii from 5. I'll try it out and see how it goes. Thanks!