HELP-find the volume of the dish
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HELP-find the volume of the dish

[From: ] [author: ] [Date: 11-08-09] [Hit: ]
We will integrate by disks.(Could go by shells, too, doesnt matter.But what is the radius?At z=0,......
A PIE WITH CIRCULAR CROSS SECTIONS IS 24 CM ACROSS THE TOP , 16 CM ACROSS THE BOTTOM, AND 6 CM DEEP. FIND THE VOLUME OF THIS DISH? Solve using integrals

Can some one please help me with this question? I am very much lost =(

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y = 3/2 x - 12 <== represents outer side of pie
y = 6 <== top of pie
y = 0 <== bottom of pie
x = 0 <== center

Volume = Area bounded by the above four equations rotated about the y axis.

V = π integral (a to b) [ f(y)^2 ] dy

y = 3/2 x - 12

x = 2/3 y + 8

V = π * integral (0 to 6) [ (2/3 y + 8)^2 ] dy <=== integral to evaluate

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Insufficient information without a picture. Are the sides of the pie straight segments?

Assuming they are, let's use cylindrical coordinates (p, theta, z), and center the pie on the origin, resting on top of the z=0 plane. We will integrate by disks. (Could go by shells, too, doesn't matter.)

You can set up the triple integral for

p dp d(theta) dz

where the limits are p=0 to radius
theta goes from 0 to 2 pi (all the way around)
z goes from 0 to 6

But what is the radius?
At z=0, p (rho, the radius) is 8. at z=6, p=12. You can derive the formula that p = 8 + 2z/3
so the limits of integration for p are 0 and 8 + 2z/3

Is that enough, can you do the integrals from there?

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You can rotate a line around the x axis to get a pie. Locate (0,8) on the y axis, and (6,12).

Connect these two points and rotate this segment around the x axis.
This solid will be a truncated cone, the left end will be a circle with diameter 16, and the right end will be a circle with diameter 24.
The distance between them is the height, 6.

Find the volume by disk method.

The line would be y= (2/3)x+8 or (2x+24)/3

Integrate [ pi(2x+24)^2dx from 0 to 6= 608pi


:) R

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the pie can be considered as a 6 cm. top slice of an inverted cone.
from simple geometry the depth of the total cone is 18 cm.
volume of total cone = 1/3 x pi x 12^2 x 18 = V...............solve for V
volume of 'total cone minus the slice' = 1/3 x pi x 8^2 x 12 = v.............solve for v
hence, volume of pie = (V - v ).................solve.
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