calculus ( can you explain)?

Question

Chris has found a tub in which to stand a christmas tree.The outline of the container is the curve y= 1/20 x^2 - 20 for 0

Answer 1

Sure. It always helps to draw out what you are trying to do so first look at a plot of the curve. It goes from (30,25) down to the x-axis at (20,0) , then down to (0,-20) for the minimum before going back up (-20,0) and finally ending at (-30,25). Now the bottom of the tub is y=0 so the volume between the x values of -20 and 20 is just the volume of a cylinder or pi*H*R^2. and H=25 and R=20.

For the remaining volume we must integrate. The element of volume is an annular region that is the circumference at some distance R (this is 2*pi*R) from the center of the tub times the Height (H) and times the infinitesimal thickness of the region (dx). This is:

dV = 2*pi*R*H*dx with H=25 - y and R=x. This must then be integrated from 20 to 30.
I do not know an easy way to type the integral symbol in a text box so I will leave that to you. But the thing to be integrated is:

2*pi*x*(25 - y)*dx with y= (1/20) x^2 - 20

2*pi*x(45 - x^2/20)dx

Integrated this is: 2*pi*[ (45/2)x^2 - (1/80)x^4] evaluated from 20 to 30.

This can be changed to: [ pi*x^2 ]*[ 45 - (1/40)x^2 ]

The rest is just math and I did that by hand and got 16,250*pi square cm. This looks OK.

To check, calculate the approximate volume by using a straight line instead of a curved one between 20 and 30 . The circular volume inside x=20 is 10,000*pi and the volume inside x=30 is 22,500*pi. Half of the difference between these is 6,250*pi and added to the 10,000*pi gives 16,250*pi. Seems odd that this is exactly what was integrated to so I must be in error somewhere. But I don't see where. You can also integrate using a disc parallel to the x-axis and integrating from y= 0 to y=25.

The thing to be integrate is the area of a disc times the thickness. The thickness is just dy and the area is pi*x^2. so we have:

(pi*x^2)dy = pi*(20 y +400)dy

This gives volume = 10y^2 +400y evaluated from y=0 to 25
This also gives 16,250. This is actually the easier way to do the integration. Should have done it this way first.