I have a test in a couple of hours and i dont know how to do volume questions please help me out with thiss pleasee
The base of a particular solid S is a circular disk ( a "filled in circle") with radius 3. Cross sections perpendicular to the base and perpendicular to the x axis are squares. What is the volume of the solid?
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$\begingroup$We can let our circle have centre the origin. So the circle has equation $x^2+y^2=9$.
We first want to find the area $A(x)$ of cross-section "at" $x$.
This cross-section is a square with sides $2\sqrt{36-x^2}$: big when $x=0$, and small when $x$ is close to $3$ or $-3$. It is $2\sqrt{36-x^2}$ because the cross section extends from the upper boundary of the circle at $x$ to the lower boundary of the circle at $x$. So the area of cross-section is $4(36-x^2)$. Thus our volume is $$\int_{-3}^3 (36-4x^2)\,dx.$$ It is convenient to exploit the symmetry, and instead calculate $$2\int_0^3 (36-4x^2)\,dx.$$ This makes the calculation a little simpler.
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