How do you find the volume of the solid obtained by rotating the region bounded by the curves #y^2=4x#, #x=0# and #y=4# about the y axis?

Answer 1

#32pi#

The graph (before rotation) is shown below graph{sqrt(4x) [-0.5, 4.5, -0.5, 4.5]}

Slice the solid generated in a manner that is normal to the #x# axis, each slice having a thickness of #"d"x#.

Each slice has volume of

#"d"V = pi y^2 "d"x#
#= 4pix "d"x#

The total volume is given by

#int_0^4 4pix "d"x = [2pix^2]_0^4#
#=32pi#
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 volume of the solid obtained by rotating the region bounded by the curves ( y^2 = 4x ), ( x = 0 ), and ( y = 4 ) about the y-axis, you can use the method of cylindrical shells.

The formula to find the volume using cylindrical shells is:

[ V = 2\pi \int_a^b xf(x) , dx ]

In this case, the bounds of integration are from ( x = 0 ) to ( x = 4 ), as these are the x-coordinates where the curves intersect.

First, solve ( y^2 = 4x ) for ( x ) to get ( x = \frac{y^2}{4} ). Then, the equation ( x = 4 ) represents the vertical line.

Now, the integral becomes:

[ V = 2\pi \int_0^4 x\sqrt{4x} , dx ]

Simplify the integral:

[ V = 2\pi \int_0^4 2x^{3/2} , dx ]

[ V = 4\pi \int_0^4 x^{3/2} , dx ]

Integrate ( x^{3/2} ) with respect to ( x ):

[ V = 4\pi \left[ \frac{2}{5} x^{5/2} \right]_0^4 ]

[ V = \frac{8\pi}{5} (4^{5/2} - 0) ]

[ V = \frac{8\pi}{5} (32) ]

[ V = \frac{256\pi}{5} ]

So, the volume of the solid obtained by rotating the region bounded by the curves ( y^2 = 4x ), ( x = 0 ), and ( y = 4 ) about the y-axis is ( \frac{256\pi}{5} ).

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