Rotation area of a function

Question

Im faced with the following problem.

Calculate the area of the "cone" that is created when the function

$$y = \frac{x^2}{32}$$

$$0≤y≤24$$ rotates around the y-axis.

If we draw the cone that is created we can see that the hight will be 24. However we have to change our function.

$$y = \frac{x^2}{32} \implies \sqrt{32y} = x = f(y)$$

The formula for the rotation area is the following:

$$Area = 2\pi\cdot\int f(x)\cdot \sqrt{1+f'(x)}dx $$

I've tried to go continue from this however, i cant manage the integral part. Can any1 help me?

/John

Answer 1

For the area of the surface of revolution the set up should be for $x=f(y)=\sqrt{32y}$

$$A = 2\pi\cdot\int_0^{24} f(y)\cdot \sqrt{1+[f'(y)]^2}dy= 2\pi\cdot\int_0^{24} \sqrt{32y}\cdot \sqrt{1+\left[\frac12\frac{32}{\sqrt{32y}}\right]^2}dy=\\=2\pi\cdot4\cdot\int_0^{24} \sqrt{2y}\cdot \sqrt{1+\left(\frac14\frac{32}{y}\right)}dy= 8\pi\cdot\int_0^{24} \sqrt{2y+16}dy=8\pi\left[\frac13(2y+16)^{\frac32}\right]_0^{24}$$

Answer 2

Your formula is missing a square in the derivative.

The rotational area is given by

$$ 2\pi \int x ds = 2\pi \int x \sqrt{dx^2 + dy^2} $$

where $ds$ is the arc length element of the curve $y = \dfrac{x^2}{32}$

If you choose to integrate w.r.t $y$, the integral looks like this

\begin{align} 2\pi \int x \sqrt{dx^2 + dy^2} &= 2\pi \int_0^{24} x(y)\sqrt{1+\left(\frac{dx}{dy}\right)^2}dy \\ &= 2\pi \int_0^{24}4\sqrt{2y} \sqrt{1 + \left(\frac{2\sqrt{2}}{\sqrt{y}}\right)^2} dy \\ &= 8\pi\sqrt{2} \int_0^{24} \sqrt{y} \sqrt{1+\frac{8}{y}} dy \\ &= 8\pi\sqrt{2} \int_0^{24} \sqrt{y+8} \ dy \\ &= 8\pi\sqrt{2}\cdot \frac{2}{3} (y+8)^{3/2}\Bigg|_0^{24} \end{align}