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Math Help - Evaluating Integrals

  1. #1
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    Evaluating Integrals

    Evaluate the following integral by interpeting it in terms of areas:

    The definite integral of f(x) from 1 to 0

    f(x) = sqrt(1 - x^2)dx = (1/4)3.14x^2 = (1/4)3.14(1)^2 = 3.14/4

    This is an example that was given in my text book. I don't undersatnd the second step (that part in bold type). Can someone please exlpain why the anti-derivative of sqrt(1 - x^2)dx is (1/4)3.14x^2?
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  2. #2
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by zachb View Post
    Evaluate the following integral by interpeting it in terms of areas:

    The definite integral of f(x) from 1 to 0

    f(x) = sqrt(1 - x^2)dx = (1/4)3.14x^2 = (1/4)3.14(1)^2 = 3.14/4

    This is an example that was given in my text book. I don't undersatnd the second step (that part in bold type). Can someone please exlpain why the anti-derivative of sqrt(1 - x^2)dx is (1/4)3.14x^2?
    They cheated! But it works.

    Notice that f(x) = sqrt(1 - x^2) is the equation of the upper half of a circle with radius 1. Notice that if you integrate from x = 0 to x = 1, you are finding the area of (1/4) of the area of that circle. Let's cheat and do this the easy way:

    (1/4)*A(rea of a circle) = (1/4)*pi*r^2 = (1/4)*3.14*1 = 3.14/4
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by zachb View Post
    Evaluate the following integral by interpeting it in terms of areas:

    The definite integral of f(x) from 1 to 0

    f(x) = sqrt(1 - x^2)dx = (1/4)3.14x^2 = (1/4)3.14(1)^2 = 3.14/4

    This is an example that was given in my text book. I don't undersatnd the second step (that part in bold type). Can someone please exlpain why the anti-derivative of sqrt(1 - x^2)dx is (1/4)3.14x^2?
    I got a different answer:

    EDIT: Ah, okay, ecMathGeek to the rescue! (Why do I have to think so complicated all the time?). I'll leave this up though, it might help somebody
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  4. #4
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    Here is another way. If you do not like it then do not use it.
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