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What is the net area between #f(x) = cscx -cosxsinx# and the x-axis over #x in [pi/6, (5pi)/8 ]#?

Answer 1

Charles Anctil

#int_(pi/6)^((5pi)/8)cscx-cosxsinxdxapprox1.4181#

We want to find #I=int_(pi/6)^((5pi)/8) cscx-cosxsinxdx#.

We do this by integrating term-by-term and then applying the limits.

#intcscxdx=-ln(cscx+cotx)#

This integral is well known

#int-cosxsinxdx#

Let #u=cosx# and #du=-sinxdx#

Then #int-cosxsinxdx=intudu=1/2u^2=1/2cos^2x#

So #I=[1/2cos^2x-ln(cscx+cotx)]_(pi/6)^((5pi)/8)approx1.4184#

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Answer 2

Charles Arocha

The net area between ( f(x) = \csc(x) - \cos(x)\sin(x) ) and the x-axis over ( x ) in ( \left[\frac{\pi}{6}, \frac{5\pi}{8}\right] ) is given by the definite integral ( \int_{\frac{\pi}{6}}^{\frac{5\pi}{8}} |\csc(x) - \cos(x)\sin(x)| , dx ). Since the integrand involves absolute value, we need to consider the intervals where ( \csc(x) - \cos(x)\sin(x) ) is positive and negative separately. This integral can be challenging to compute directly, and you may need to use numerical methods or software to obtain a numerical value.

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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.