Hello, Everyone!! Its been a while, but I have decided to start a series on Calculus topics, especially since I will be taking Cal BC this coming school year, I thought it would be cool to refresh these topics in my mind and give you some info on how Calculus can help you out. Really. I may not be in Calculus already, but I have been exposed to many calculus related problems during my ninth and tenth school years in Canada. Calculus is so amazing, and broad but at the same time, can be used to find volumes of solids of various different kinds. So, don't worry if you don't know the formula for finding the volume of a sphere during a test or quiz. Calculus has got you.
Lets do an example together. Let's find the volume of a sphere (the midpoint being 0) from the values of a and -a. lets first find the graphic formula for the area of the cross-sectional surface: x^2 + r^2 = a^2 <--- finding area through the circle formula - step 1 r = √(a^2 - x^2) <--- solving for r - step 2 S(x) = π(a^2 - x^2), -a ≤ x ≤ a <--- finding area of the circle, domain - step 3 V = ∫(from a to -a) [π(a^2 - x^2) dx] <--- set volume to definite integral - step 4 = π[ a^2x - 1/3x^3](from a to -a) <--- solve - step 5 = π[4/3a^3] = 4/3a^3π So, we finally derived the volume formula of a sphere, and since it is a generalized formula, a can be any value (radius). Good luck,
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