# Determining the Surface Area of a Solid of Revolution

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M8-9: surface are of surfaces of revolution

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• If a surface is obtained by rotating about the x-axis from $t = a$ to $b$ the curve of the parametric equation

$\left\{\begin{matrix}x = x \left(t\right) \\ y = y \left(t\right)\end{matrix}\right.$,

then its surface area A can be found by

$A = 2 \pi {\int}_{a}^{b} y \left(t\right) \sqrt{x ' \left(t\right) + y ' \left(t\right)} \mathrm{dt}$

If the same curve is rotated about the y-axis, then

$A = 2 \pi {\int}_{a}^{b} x \left(t\right) \sqrt{x ' \left(t\right) + y ' \left(t\right)} \mathrm{dt}$

I hope that this was helpful.

• The formula ti evaluate the surface area of a function $y \left(x\right)$ rotated around the $x$ axis is:

 S=2piinty(x)dl

where $\mathrm{dl} = \sqrt{{\mathrm{dx}}^{2} + {\mathrm{dy}}^{2}}$. If we express all in the $x$ variable running in some interval $\left[a , b\right]$ we have then:

 S=int_a^by(x)sqrt(1+(dy/dx)^2)dx

In our case is very easy to verify that $y \left(x\right) = \frac{2}{x}$ and the interval on which $x$ runs is $\left[x \left(1\right) , x \left(5\right)\right] = \left[2 , 50\right]$.

So in order to evaluate this surface we have to solve the integral:

 S=int_2^(50)2/xsqrt(1+4/x^4)dx

From now on the solution is quite long, so I will show you just the basic steps:

• Multiply numerator and denominator by ${x}^{2}$

$\implies S = {\int}_{2}^{50} 2 \frac{\sqrt{4 + {x}^{4}}}{x} ^ 3 = {\int}_{2}^{50} 4 \frac{\sqrt{1 + {x}^{4} / 4}}{x} ^ 3$

• Perform the change of variable ${x}^{2} / 2 = \sinh \left(z\right)$

$\implies {\int}_{{z}_{1}}^{{z}_{2}} \frac{{\cosh}^{2} z}{{\sinh}^{2} z} \mathrm{dz} = {\int}_{{z}_{1}}^{{z}_{2}} \left({\coth}^{2} z\right) \mathrm{dz}$

where ${z}_{1} = {\sinh}^{-} 1 \left(2\right)$ and ${z}_{2} = {\sinh}^{-} 1 \left(1250\right)$

• Find the antiderivative of ${\coth}^{2} \left(z\right) = 1 + {\left(\frac{1}{\sinh} z\right)}^{2}$
• You shold end up whith the expression:

 S=(z-cothz)|_(z_1)^(z_2)

which finally gives:

 S~=6.498