Choosing the Right Variable for Radius of Shell in Cylindrical Shell Method

Jon1436
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Im new to finding volume using the cylindicral shell method so what should i do. I know I will eventually plug equations into the integral 2piX(f(x)-g(x))dx

x=y x+2y=3 and y=0 revolve about the x axis
 
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Draw a picture of the region. Now visualize what the shells are going to look like around the region. Then take your formula and think of it as 2*pi*(radius of shell)*(length of shell)*d(radius of shell). If you are rotating around the x-axis, which variable is a good choice for radius of shell. x or y?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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