What is the surface area produced by rotating #f(x)=2/(e^x-3), x in [0,2]# around the x-axis?
See the answer below:
By signing up, you agree to our Terms of Service and Privacy Policy
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.
By signing up, you agree to our Terms of Service and Privacy Policy
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.
- Let R be the region enclosed by #f(x) = sinx, g(x) =1-x, and x=0#. What is the volume of the solid produced by revolving R around the x-axis?
- a) Show that the formula for the surface area of a sphere with radius #r# is #4pir^2#. b) If a portion of the sphere is removed to form a spherical cap of height #h# then then show the curved surface area is #2pihr^2#?
- What is the arclength of #f(x)=sqrt(4-x^2) # in the interval #[-2,2]#?
- What is the arc length of #f(x)=x^2/12 + x^(-1)# on #x in [2,3]#?
- What is the general solution of the differential equation ? # sec^2y dy/dx+tany=x^3 #

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7