What is the surface area produced by rotating #f(x)=tanx-cos^2x, x in [0,pi/4]# around the x-axis?

Answer 1
About #1.483pi# #"u"^2#...
For revolutions around the #x#-axis, the surface area is given by:
#S = 2pi int_(alpha)^(beta) f(x) sqrt(1 + ((dy)/(dx))^2) \ dx#

Clearly, this is most suitable for very, very simple functions, and this is not one of those. Anyways, we should take the derivative and then square it.

#(dy)/(dx) = sec^2x + 2sinxcosx#
#=> ((dy)/(dx))^2 = sec^4x + 4sinxcosxsec^2x + 4sin^2xcos^2x#

So, the surface area integral becomes:

#S = 2pi int_0^(pi/4) (tanx - cos^2x)sqrt(1 + sec^4x + 4tanx + sin^2 2x) \ dx#

This is evidently a time sink to solve, so I will just plug it into Wolfram Alpha to evaluate like that. We then get:

#color(blue)(S ~~ 1.483pi)# #color(blue)("u"^2)#
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Answer 2

To find the surface area produced by rotating the function ( f(x) = \tan(x) - \cos^2(x) ) around the x-axis over the interval ( x \in [0, \frac{\pi}{4}] ), we use the formula for surface area of revolution:

[ S = \int_{a}^{b} 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

Where:

  • ( y = f(x) )
  • ( \frac{dy}{dx} ) is the derivative of ( f(x) )

First, we find the derivative of ( f(x) ):

[ f'(x) = \sec^2(x) + 2\cos(x)\sin(x) ]

Next, we plug in the expressions for ( f(x) ) and ( f'(x) ) into the surface area formula and integrate over the given interval:

[ S = \int_{0}^{\frac{\pi}{4}} 2\pi (\tan(x) - \cos^2(x)) \sqrt{1 + (\sec^2(x) + 2\cos(x)\sin(x))^2} , dx ]

This integral represents the surface area of revolution for the function over the specified interval. Evaluating this integral will give the surface area.

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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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