What is the net area between #f(x) = x-sin^2x # and the x-axis over #x in [0, 3pi ]#?

Answer 1

The area is #=(3pi)/2(3pi-1)#

The area is #int_0^(3pi)(x-sin^2x)dx# #cos2x=1-2sin^2x# #:. sin^2x=(1-cos2x)/2# The area is #int_0^(3pi)(x-sin^2x)dx=int_0^(3pi)(x-1/2+cos(2x)/2)dx#
#=[x^2/2-x/2+sin(2x)/4]_0^(3pi) # #=((9pi^2)/2-(3pi)/2+sin (6pi)/4)-(0)#
#=(9pi^2)/2-(3pi)/2=(3pi)/2(3pi-1)#
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Answer 2

To find the net area between ( f(x) = x - \sin^2(x) ) and the x-axis over ( x ) in ( [0, 3\pi] ), you integrate the absolute value of ( f(x) ) from ( x = 0 ) to ( x = 3\pi ). This accounts for both positive and negative regions of the curve. Therefore, the net area is the integral of ( |x - \sin^2(x)| ) from ( 0 ) to ( 3\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.

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.

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.

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.

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