Find the areas of surfaces of revolution and Work

[shivers20]

I am doing some practice problems for my exam tomorrow and I am having a bit of trouble when it comes to certain steps.

I need to find the area of surface of revolution. I need to just write down the expression whatever that means.

1. Find the area of the surface generated by revolving the curve y = x^3, 0 < x < 2, about the axis.

Here goes:

a = 0, b = 1, y = x^3, dy/dx = 3x^2

Sqrt 1 + (3x^2)^2

= 1 + 3x^2

S= {1   2TT (x^3) (1+3x^2)dx = 2TT ? 
   }0

This is where I became very confused.

2) If a force of 90 N stretches a spring 1m beyond its natural length, how much work does it take to stretch the spring 5m beyond its natural length?

I don't know what F(x) is.

W= {5   90xdx=  ?x^2dx |5       90/4?
   }1                  |1

______________________
Edited by stapel -- Reason for edit: Attempting to display multi-line formatting.

 

[stapel]

Sqrt 1 + (3x^2)^2 = 1 + 3x^2

Where does the square root come from? How do you get that sqrt[1] + (3x²)² equals 1 + 3x²?

In your formatted bit, what does "S" stand for? Where did the "T" variable come from? Why do you write "TT" instead of "T²"? What are the curly-braces (extended over two lines) meant to indicate? (It is usually better to use standard formatting, in a single line, as is explained here.)

I don't know what F(x) is.

I don't see any function F(x) in the statement of the exercise. Where did this come from? What do you mean by the multi-line formatted part?

Please reply with clarification. Thank you.

Eliz.

 

[galactus]

I am doing some practice problems for my exam tomorrow and I am having a bit of trouble when it comes to certain steps.

I need to find the area of surface of revolution. I need to just write down the expression whatever that means.

1. Find the area of the surface generated by revolving the curve y = x^3, 0 < x < 2, about the axis.

Here goes:

a = 0, b = 1, y = x^3, dy/dx = 3x^2

Sqrt 1 + (3x^2)^2

= 1 + 3x^2

S= {1   2TT (x^3) (1+3x^2)dx = 2TT ? 
   }0

This is where I became very confused.

You're confused. What about us. I just figured out your TT is supposed to represent pi?.

They stated to just set up the integral, but don't solve. Lucky you.

\(\displaystyle \sqrt{1+(3x^{2})^{2}}\) does not equal \(\displaystyle 1+3x^{2}\)

Also, you've changed your limits of integration from [0,2] to [0,1].

\(\displaystyle (3x^{2})^{2}=9x^{4}\)


2) If a force of 90 N stretches a spring 1m beyond its natural length, how much work does it take to stretch the spring 5m beyond its natural length?

I don't know what F(x) is.

F(x) is Hooke's law, where F(x) = kx.

F(1)=k=90 N/m

\(\displaystyle W=\int_{0}^{5}90xdx\)

\(\displaystyle W=45x^{2}|_{0}^{5}\)

W= {5   90xdx=  ?x^2dx |5       90/4?
   }1                  |1

______________________
Edited by stapel -- Reason for edit: Attempting to display multi-line formatting.