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Math Help - area of a circle (integral)

  1. #1
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    area of a circle (integral)

    use a double integral and trigonometric substitution to find the area of a circle with radius r. use the formula int(cos^2u)du = (1/2)u + (1/4)sin2u + C

    so ive gotten this far(split the integral into a quarter, then will multiply by 4)...

    4*int(0 to r)*int(0 to sqr[r^2-x^2]) 1 dydx

    integrating sqr[r^2-x^2] seems to get messy, so i am assuming my substitution comes in here. but im not sure what it would be
    would it be something like cos[sqr(r^2-x^2)/r] ?

    there are also 3 other questions, which want me to find the volume of a sphere using triple integrals, and hypervolume of a hypersphere using quadruple integrals, and the volume of a hypersphere using n integrals. however, i think before i start those, i need to figure out the first basic one, but i cant.
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  2. #2
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    For the first one you have \int_0^r \sqrt{r^2-x^2}dx right? How about if you let x=r\sin(\theta). Can you show this equates to r^2\int_0^{\pi/2} \cos^2(\theta)d\theta. You can do that right?

    Yea well, I guess you got \int_0^{r}\int_0^{\sqrt{r^2-x^2}} dydx. Same dif right?
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  3. #3
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    same question

    Quote Originally Posted by chrisc View Post
    use a double integral and trigonometric substitution to find the area of a circle with radius r. use the formula int(cos^2u)du = (1/2)u + (1/4)sin2u + C

    so ive gotten this far(split the integral into a quarter, then will multiply by 4)...

    4*int(0 to r)*int(0 to sqr[r^2-x^2]) 1 dydx

    integrating sqr[r^2-x^2] seems to get messy, so i am assuming my substitution comes in here. but im not sure what it would be
    would it be something like cos[sqr(r^2-x^2)/r] ?

    there are also 3 other questions, which want me to find the volume of a sphere using triple integrals, and hypervolume of a hypersphere using quadruple integrals, and the volume of a hypersphere using n integrals. however, i think before i start those, i need to figure out the first basic one, but i cant.

    I have the same question as you did, but I dont know how to get the rest. Did you figure it out?
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