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In integration such as, $$\int_{a}^{b} f(x) dx$$ It is said that that is the exact same as: $$\int_{a}^{b} f(y) dy = \int_{a}^{b} f(\theta) d\theta = \int_{a}^{b} f(\alpha) d\alpha = \int_{a}^{b} f(\psi)d\psi$$ An the list goes on forever. I cannot understand, How is it justified to...

I asked the same thing on Physicsforums, I am hoping for some feedback here as well. I saw this method of calculating: \displaystyle{I = \int_{0}^{1} \log^2(1-x)\log^2(x) dx} http://math.stackexchange.com/questions/959701/evaluate-int1-0-log21-x-log2x-dx Can you take a look at M.N.C.E.'s...

HI there, Take a question like, (Q#) Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet.My question is, WHY do you assume a hole doesnt exist from which air leaves the balloon? Also think about...