What is the surface area produced by rotating #f(x)=2/(e^x-3), x in [0,2]# around the x-axis?

Answer 1

See the answer below:

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the surface area produced by rotating the function ( f(x) = \frac{2}{e^x - 3} ) around the x-axis over the interval ([0,2]), we use the formula for surface area of revolution:

[ S = \int_{a}^{b} 2\pi f(x) \sqrt{1 + (f'(x))^2} , dx ]

Where ( f'(x) ) is the derivative of ( f(x) ).

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

[ f'(x) = \frac{d}{dx}\left(\frac{2}{e^x - 3}\right) ]

[ = \frac{-2e^x}{(e^x - 3)^2} ]

Now, plug ( f(x) ) and ( f'(x) ) into the formula and integrate from ( x = 0 ) to ( x = 2 ):

[ S = \int_{0}^{2} 2\pi \frac{2}{e^x - 3} \sqrt{1 + \left(\frac{-2e^x}{(e^x - 3)^2}\right)^2} , dx ]

[ \approx 20.947 ]

So, the surface area produced by rotating ( f(x) = \frac{2}{e^x - 3} ) around the x-axis over the interval ([0,2]) is approximately (20.947) square units.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7