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  1. #1
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    mass

    The question:
    A cardboard figure has the shape shown below. The region is bounded on the left by the line x = a, on the right by the line x = b, above by f(x), and below by g(x). The density p(x) in gm/ cm2 varies only with x.

    Find the integral needed for the total mass of the figure. Give your answer using the form below.

    I know that A =a and B=b as far as the limit of integration and that the area is (f(x)-g(x)) dx so mass is
    (f(x)-g(x)) dx* p(x) but im not sure how to expresses it as H(x), and anyone please help. Thank you
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  2. #2
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    Quote Originally Posted by acosta0809 View Post
    The question:
    A cardboard figure has the shape shown below. The region is bounded on the left by the line x = a, on the right by the line x = b, above by f(x), and below by g(x). The density p(x) in gm/ cm2 varies only with x.

    Find the integral needed for the total mass of the figure. Give your answer using the form below.

    I know that A =a and B=b as far as the limit of integration and that the area is (f(x)-g(x)) dx so mass is
    (f(x)-g(x)) dx* p(x) but im not sure how to expresses it as H(x), and anyone please help. Thank you
    H(x) = \rho(x)[f(x)-g(x)]dx

    what else do you think H(x) could be?
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  3. #3
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by skeeter View Post
    H(x) = \rho(x)[f(x)-g(x)]dx

    what else do you think H(x) could be?
    Is it wrong to write \int_a^b \rho(x)[f(x)-g(x)]dx = \int_{g(x)}^{f(x)} \int_a^b \rho (x) dxdy?
    Or should I write \int_a^b \rho(x)[f(x)-g(x)]dx =\int_a^b \int_{g(x)}^{f(x)}  \rho (x) dydx? Or it's exactly the same thanks to Fubini?
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  4. #4
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    Quote Originally Posted by arbolis View Post
    Is it wrong to write \int_a^b \rho(x)[f(x)-g(x)]dx = \int_{g(x)}^{f(x)} \int_a^b \rho (x) dxdy?
    Or should I write \int_a^b \rho(x)[f(x)-g(x)]dx =\int_a^b \int_{g(x)}^{f(x)}  \rho (x) dydx? Or it's exactly the same thanks to Fubini?
    Been awhile since I've done double integrals, but if I remember correctly, the outside integral must have constant limits of integration.
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  5. #5
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by skeeter View Post
    Been awhile since I've done double integrals, but if I remember correctly, the outside integral must have constant limits of integration.
    Ah you're right. So \int_a^b \int_{g(x)}^{f(x)}  \rho (x) dydx
    is maybe good.
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