# How do you find the volume bounded by #y^2=x^3-3x^2+4# & the lines x=0, y=0 revolved about the x-axis?

It's this

but bounded in the first quadrant

A small element of width

And so will have volume, when revolved about x-axis, of

So we can say that

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

To find the volume bounded by the curve (y^2=x^3-3x^2+4), the x-axis, and the y-axis, when revolved about the x-axis, you can use the method of cylindrical shells. The volume can be calculated using the formula:

[ V = 2\pi \int_{a}^{b} x\cdot |y| , dx ]

where ( a ) and ( b ) are the x-values where the curve intersects the x-axis. In this case, ( a ) and ( b ) are the x-intercepts of the curve. To find these intercepts, set ( y = 0 ) and solve for ( x ) in the equation ( y^2 = x^3 - 3x^2 + 4 ).

Once you find ( a ) and ( b ), integrate ( x \cdot |y| ) with respect to ( x ) from ( a ) to ( b ) and multiply by ( 2\pi ).

[ V = 2\pi \int_{a}^{b} x\cdot |y| , dx ]

[ V = 2\pi \int_{a}^{b} x \sqrt{x^3-3x^2+4} , dx ]

Evaluate this integral to find the volume.

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

- How do you find volume by rotating area enclosed by #y=x^3# and #y=sqrt(x)# about x=1?
- What is the general solution of the differential equation ? # 2xdy/dx = 10x^3y^5+y #
- What is the value of #F'(x)# if #F(x) = int_0^sinxsqrt(t)dt# ?
- How do you sketch the slope field for the differential equation #1/2 x +y -1#?
- #f:RR->RR;f(x)=e^x-x-1;g:[0,1]->RR;g(x)=f(x)+x#.How to calculate the volume of the body obtained by rotating the graph of the function "g" axis OX?

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