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Math Help - Volume of a different solid - ie next question

  1. #1
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    Volume of a different solid - ie next question

    Hi again

    I have another trig function so not sure what I am doing

    using the V= (definite integral "S" a b) 2pi x f(x) dx formula

    V = 2pi S a = 0; b=pi^1/2 sin (x^2) dx

    then I am not sure what sin (x^2) dx becomes - is it = -(1/3)cos(x^3)???

    I feel very confused

    Thanks

    Calculus beginner
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  2. #2
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    Quote Originally Posted by calcbeg View Post
    Hi again

    I have another trig function so not sure what I am doing

    using the V= (definite integral "S" a b) 2pi x f(x) dx formula

    V = 2pi S a = 0; b=pi^1/2 sin (x^2) dx

    then I am not sure what sin (x^2) dx becomes - is it = -(1/3)cos(x^3)???

    I feel very confused

    Thanks

    Calculus beginner
    No. Here you'll need the trig identity

     <br />
\sin^2x = \frac{1- \cos 2x}{2}<br />
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  3. #3
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    next step

    So is sin (x^2) = (1- cos 2x)/2 the integral or do I still need to do that?

    Is (1-cos 2x)/2 the antiderivative of sin x^2??

    Thanks

    Calculus beginner
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  4. #4
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    Quote Originally Posted by calcbeg View Post
    So is sin (x^2) = (1- cos 2x)/2 the integral or do I still need to do that?

    Is (1-cos 2x)/2 the antiderivative of sin x^2??

    Thanks

    Calculus beginner
    No - it's an identity. You might recognize the following double angle formulas

     <br />
\begin{array}{ll}<br />
\cos 2x &= \cos^2 x - \sin^2x\\<br />
&= 2 \cos^2x - 1\\<br />
&= 1 - 2 \sin^2x\\<br />
\end{array} <br />
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