Question

Write a Riemann sum and then a definite integral representing the volume of the hemisphere, using the slice shown in the figure below where d = 18 mm. Evaluate the integral exactly. Use your work to answer the questions below.

What is the approximate volume of the slice with respect to y?

Answer 1

the figure?

Answer 2

I don't think you need the figure...a hemisphere with a diameter of 18mm

Answer 3

http://www.webassign.net/hgmcalc/8-1-13alt.gif

Answer 4

@stamp http://www.webassign.net/hgmcalc/8-1-13alt.gif

Answer 5

At height y, x can be calculated by sqrt(81-y²), because the radius is 9.
So at height y, there is a disk with radius sqrt(81-y²) and thickness delta y
The volume of that disk is pi*x²*delta y, so for the Riemann-sum you've got something like:\[\sum_{k=0}^{n}\pi (81-y _{k}^2)\Delta y\]if you have n+1 disks.
You can probably do the conversion to an integral yourself...