(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Say the sphere of radius "a" is made out of various rings with height R(x) and thickness dx. Adding up all of the rings will form a sphere, and in order to do that, I have to integrate.

2. Relevant equations

Trigonometric Substitution:

[tex]\frac{x}{a}=sin \theta[/tex]

[tex]dx=a cos \theta d\theta[/tex]

Function of ring height related to position in cartesian plane:

[tex]R(x)=\sqrt(a^2-x^2)[/tex]

3. The attempt at a solution

Set up the integral, I just want to make my life simpler and integrate half a circle:

[tex]A=\int_0^a 2\pi R(x) dx[/tex]

Substitute R(x):

[tex]\int_0^a 2\pi \sqrt(a^2-x^2) dx[/tex]

Now using trigonometric substitution and factoring out a:

[tex]\int_0^a 2\pi a \sqrt(1-sin^2 \theta) dx[/tex]

Using pythagorean trig identity and trig substituting for dx:

[tex]\int_0^a 2\pi a cos\theta (a cos\theta d\theta)[/tex]

[tex]\int_0^a 2\pi a^2 cos^2\theta d\theta[/tex]

Putting out all the constants, and integrating, using tables of integral, I get:

[tex]2\pi a^2\int_0^a cos^2\theta d\theta[/tex]

[tex]2\pi a^2\int_0^a cos^2\theta d\theta[/tex]

[tex]2\pi a^2 (\frac{\theta}{2}+\frac{sin(2\theta)}{4})|_0^a[/tex]

The problem is, how do I solve the final part of the definite integral? What I know is that the parenthesis should be equal to 1 because the surface area is 4 pi r^2 and I integrated half a sphere.

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# Homework Help: Surface Area of a Sphere by Integration

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