Recent content by Chenkel

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    B Question about change of variables

    So ##\int_a^b f(u(x))u'(x)dx = \int_{u(a)}^{u(b)} f(t)dt## is still true when ##u(a) = u(b)##?
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    B Question about change of variables

    The fact that the proof you provided for the change of variables definite integration equation doesn't work with a non-injective ##u## function doesn't diminish the validity of the proof itself, right?
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    B Question about change of variables

    I agree that the left hand side of that equation doesn't equal the right side, and I think I see the general problem, if ##u=\sin \theta## and the limits of integration are ##0## to ##2\pi## then the result will be null in the change of variables formula regardless of what the original function is.
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    B Question about change of variables

    I'm not sure what you mean by somethings going wrong there. The integral ##\int_0^{2\pi} \sin \theta \ d\theta## is 0 using the fundamental theorem of calculus ##[-cos(\theta)]\Bigg|_0^{2\pi}=-\cos(2\pi) + cos(0) = 0## and the right hand side of the equation you posted is the integral from...
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    B Question about change of variables

    Interesting, thank you for the insight!
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    B Question about change of variables

    I think I probably said it right, I hope so.
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    B Question about change of variables

    Now I'm a little confused and not sure of I said that right
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    B Question about change of variables

    So perhaps the reason my proof worked is because when I let ##u=r\cos(\theta)## I kept ##\theta## in the interval ##[0, \pi]## and in that interval u is one-to-one and and invertible if I'm not mistaken.
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    B Question about change of variables

    What do we do if ##a## and ##b## are largely separated and ##u## is a periodic function that oscillates more than once in the interval ##[a, b]##? For example I chose ##u=r\cos(\theta)## and I chose ##\theta = [\pi, 0]## and I got the correct answer for the area of the circle, but if I chose...
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    B Question about the fundamental theorem of calculus

    Using what you wrote I'm able to see the how the left approach derivative of the accumulation function with the mean value theorem is equal to the function in the integrand. Thank you for the insight!
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    B Question about the fundamental theorem of calculus

    Hello everyone, I've been brushing up on some calculus and had some new questions come to mind. I notice that most proofs of the fundamental theorem of calculus (the one stating the derivative of the accumulation function of f is equal to f itself) only use a limit where the derivative is...
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    B Question about units for "area under curve"

    Hello everyone, I am curious, suppose you have a function ##f(x)=x^3## and you to find the area under the curve from 0 to x, the area would be ##\frac {x^4}{4}## but this is units of ##L^4## if x is length, but area is units of ##L^2## so what is going on here? The reason I'm curious is I...
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    B Question about change of variables

    So just to prove my formulas are consistent with the change of variables formula you proved. ##f(x) = \sqrt{r^2 - x^2}## ##u(\theta) = r\cos(\theta)## ##u'(\theta) = -r\sin(\theta)## If you expand the following equation with the previous variables it's the same equation I used in my proof...
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    B Question about change of variables

    Isn't the antiderivatie of ##\frac {1}{\sqrt{t}}## equal to ##2\sqrt{t}## not ##\frac {-2} {3} t^{\frac {-3}{2}}##? Thanks again for the result of the proof, I'm going to study it some more today but I think I mostly understood it.
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    B Question about change of variables

    So I believe I've been grocking the integration process more, in the first integral we are summing vertical strips with respect to x and then when we change the variable of integration to ##\theta## by reexpressing the differential area with a change in variables for the differential element...
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