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What is the net area between #f(x)=xsinx# in #x in[0,4pi] # and the x-axis?

Answer 1

Aurora Alvarado

#= - 4 pi#

graph{x sin x [-36.5, 36.48, -18.24, 18.26]}

you want the net area so you don't mind that the positive and negative values net off. you need to know, therefore

#int_0^{4 pi} x sinx dx#

we can do this by IBP using the idea that #int u v' dx= uv - int u' v dx#

#u = x, u' = 1# #v' = sin x, v = - cos x#

#int_0^{4 pi} x sinx dx# #= [- x cos x ]_0^{4 pi} - int_0^{4 pi} - cos x dx#

#= [-x cos x]_0^{4 pi} + int_0^{4 pi} cos x dx#

#= [-x cos x + sin x]_0^{4 pi}#

so we evaluate

#[- x cos x + sin x]_0^{4 pi}#

#= - 4 pi#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.