What is the surface area of the solid created by revolving #f(x) = x^2+e^x , x in [2,4]# around the x axis?

Answer 1

#V=pi1044~~3279.82#

First, take a look at the graph #f(x) = x^2+e^x, x in[2,4]#

I've drawn vertical lines at the endpoints to show where the solid would revolve around the #x#-axis. It looks like the object would ultimately resemble a circular bell.

The formula for find the volume of a shape like this is

#V=piint_("lower")^("upper")[f(x)]^2dx#

In this case, you can plug in all the values and find the integral in a straight forward fashion.

#V=pi int_2^4 [x^2+e^x]^2dx#

#V=pi (int_2^4[x^4+2x^2e^x+e^(2x)]dx)#

#V=pi [1/5x^5+2x^2e^x+4xe^x+2e^(2x)]_2^4#

#V=pi[1/5(4)^5+2(4)^2e^4+4(4)e^x+2e^(2*4)]-pi[1/5(2)^5+2(2)^2e^2+4(2)e^2+2e^(2*2)]#

#V=pi1044~~3279.82#

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 of the solid created by revolving ( f(x) = x^2 + e^x ) around the x-axis over the interval ([2, 4]):

  1. Determine the formula for the surface area of revolution, which is given by: [ SA = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

  2. Find the derivative of ( f(x) ) with respect to ( x ), which is: [ f'(x) = 2x + e^x ]

  3. Use the derivative to find the expression for ( \frac{dy}{dx} ).

  4. Substitute the expression for ( \frac{dy}{dx} ) into the formula for surface area.

  5. Integrate the resulting expression from ( x = 2 ) to ( x = 4 ) to find the surface area of the solid.

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