Volumes (Slicing) (1 Viewer)

[_ShiFTy_]

1)

Find the volume when the area bounded by y = -x^2 - 3x + 6 and x + y - 3 = 0 is rotated about x = 3


2)

The region enclosed by the parts of the line y = x/2, and x = 3, and the part of the hyperbola x^2 - y^2 = 12 is rotated about the y axis. Find the volume

 

[.ben]

for the first one i got 397/3pi units^3. it's probably wrong

 

[_ShiFTy_]

The answer for the first one is 256pi/3 units cubed

After i thought about it, do you need to split the slices into 2 sections? hmmm..shell method would be a lot easier

 

[.ben]

yes i spilt the volume into two sections. did you use the dv=pi(x^2-x^2)dy? if so what did you use ast hte second x value?

 

[_ShiFTy_]

For question one, you split it into 2 sections...the top section should look like an irregular semicircle

You should get

Δv = π [ (3 - x₁)² - (3 - x₂)² ] Δy

*but note x₁ + x₂ = - 3...eliminate either one and you should end up getting this after you change the x in terms of y

-12π ₆∫^8.25 √(33/4 - y)dy

You should get 40.5π as the answer



The second section is more fiddly:

Δv = π [ (3 - x₁)² - (3 - x₂)² ] Δy (same as before)

*but x₁ refers to y = -x + 3
x₂ refers to the parabola

After subbing in the y values, you should end up getting:

π ₂∫⁶ (y² + y + 9√(33/4 - y) - 28.5)

You should get 44.83333333π as the answer