# How do you find the volume of the solid generated by revolving the region bounded by the graphs #y=3(2-x), y=0, x=0#, about the y axis?

Volume

When

# \ \ \ \ \ V =1/3pir^2h #

# :. V = 1/3pi*2^2*6 #

# :. V = 8pi #

If you want a calculus solution, then the Volume of revolution about the

# V =int_a^b pi x^2dy#

Now:

# y=3(2-x) => y=6-3x #

# :. x=1/3(6-y) #

And so:

# V = int_0^6 pi (1/3(6-y))^2 \ dy #

# \ \ \ = 1/9 pi int_0^6 (6-y)^2 \ dy #

# \ \ \ = 1/9 pi int_0^6 (36-12y+y^2) \ dy #

# \ \ \ = 1/9 pi [36y-6y^2+1/3y^3]_0^6 #

# \ \ \ = 1/9 pi {(36*6-6*36+216/3) - (0)} #

# \ \ \ = 1/9 pi (72) #

# \ \ \ = 8 pi #

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

To find the volume of the solid generated by revolving the region bounded by the graphs ( y = 3(2 - x) ), ( y = 0 ), and ( x = 0 ) about the y-axis, you can use the method of cylindrical shells.

- Determine the limits of integration by finding the intersection points of the curves.
- Set up the integral for the volume using the formula for the volume of a cylindrical shell.
- Integrate the expression to find the volume.

The volume ( V ) can be calculated using the formula:

[ V = \int_{a}^{b} 2\pi x f(x) , dx ]

where ( f(x) ) is the height of the shell at ( x ), and ( a ) and ( b ) are the limits of integration.

In this case, the region is bounded by the y-axis and the curve ( y = 3(2 - x) ). The limits of integration are from ( x = 0 ) to ( x = 2 ), as those are the x-values where the curve intersects the y-axis.

So, the volume ( V ) is given by:

[ V = \int_{0}^{2} 2\pi x \cdot (3(2 - x)) , dx ]

You can evaluate this integral to find the volume of the solid.

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.

- How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y = 2 x^4#, y = 0, x = 1 revolved about the x=2?
- How do can you derive the equation for a circle's circumference using integration?
- How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y=sqrt(16-x^2)# and the x axis rotated about the x axis?
- What is a solution to the differential equation #dT+k(T-70)dt# with T=140 when t=0?
- What is the average value of a function #y=secx tanx# on the interval #[0,pi/3]#?

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