# Another volume of revolution around x axis

1. Jul 9, 2012

### togo

1. The problem statement, all variables and given/known data
Find the volume of the area bounded by
y^2 = 4ax
and
x=a

rotated around the x-axis.

2. Relevant equations
integral of pi R^2 dh

3. The attempt at a solution
I just don't know how to handle the x=a part of the boundary. Any hints? thanks

2. Jul 9, 2012

### SammyS

Staff Emeritus
Integrate from x=0 to x=a .

The trickier part may be understanding how to handle the integrand .

3. Jul 9, 2012

### togo

well I read lots of books on it and do lots of questions the only way to understand them sometimes is to try it and then see someone do it step by step

4. Jul 9, 2012

### SammyS

Staff Emeritus
Graphing y2 = 4ax will help for this question.

5. Jul 9, 2012

### togo

k so the integral of 4ax would be (4/2)ax^2 right?

6. Jul 9, 2012

### SammyS

Staff Emeritus
When you write
integral of pi R2 dh ,​
how is R related to y ?

7. Jul 10, 2012

### togo

R^2 and Y^2 end up being the same thing right?

8. Jul 10, 2012

### SammyS

Staff Emeritus

Yes, that is the correct integral.

What limits of integration should you use?

9. Jul 10, 2012

### togo

x=0 to x=a

but I forget how to integrate a constant (a)

10. Jul 10, 2012

### SammyS

Staff Emeritus
$\displaystyle \int\,a\,f(x)\,dx=a\,\int f(x)\,dx$

11. Jul 13, 2012

### togo

I guess that I am not sure how to do that.

for example, ∫(2)dx = 2x + C

but how does that apply here? the answer doesn't have any + in it.

12. Jul 13, 2012

### SammyS

Staff Emeritus
For a definite integral, the constant of integration has no effect, so it can be ignored.

$\displaystyle \int_{b}^{c}\,a\,f(x)\,dx=a\,\int_{b}^{c} f(x)\,dx$

In your case b=0 and c=a.

The fact that upper limit of integration is the same as the multiplicative constant is merely a coincidence.

$\displaystyle \int_{0}^{a}\,4a\,x\,dx=4a\,\int_{0}^{a} x\,dx$

13. Jul 13, 2012

### HallsofIvy

Staff Emeritus
Typically, one of the very first things you learn about integrals is that the anti-derivative of $x^n$, for n not equal to -1, is $(1/(n+1))x^{n+1}+ C$. What is that for x= 0?