# What is the surface area produced by rotating #f(x)=1/(x^2+1), x in [0,3]# around the x-axis?

Well, the surface area of a function

#S = int 2pi f(x)ds#

#= int overbrace(2pi f(x))^"Circumference"overbrace(sqrt(1 + ((df)/(dx))^2))^("Arc Length")dx#

This surface would look like:

First we get the derivative.

#(df)/(dx) = -(2x)/(x^2 + 1)^2#

And square it to get:

#((df)/(dx))^2 = (2x)^2/(x^2 + 1)^4#

So, the surface area integral is:

#S = 2pi int_(0)^(3) 1/(x^2 + 1)sqrt(1 + ((2x)^2)/(x^2 + 1)^4)dx#

#= 2pi int_(0)^(3) sqrt(1/(x^2 + 1)^2 + ((2x)^2)/(x^2 + 1)^6)dx#

There is no elementary solution to this, so we can only get the numerical integration result,

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

To find the surface area produced by rotating ( f(x) = \frac{1}{x^2+1} ), where ( x ) is in the interval ([0,3]), around the x-axis, you can use the formula for the surface area of revolution:

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

First, calculate ( f'(x) ):

[ f'(x) = -\frac{2x}{(x^2+1)^2} ]

Now, substitute the values and integrate:

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

After integrating, you'll get the surface area.

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.

- 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