let R be the region bounded by the graph of f(x)=4-x^2 and the coordinated axes in quadrant I

Question

1)sketch & label region R.

2) area of  the region R using vertical slicing.

3) area of the region R using horizontal slicing.

4)volume of  revolution  using the indicated method when region R is revolved about

a) x-axis  (discs)

b) x-axis (shells)

c) y-axis (discs)

d)y-axis (shells)

e)about x=2 (discs)

f)about x=2(shells)

g) about y=4(discs)

h) about y=4(shells)

 

 

 

Answer 1

f(x) is a parabola opening downward with vertex as (0,4) and a root at (2,0).

In the first quadrant the function can also be written in terms of y as g(y) = (4-y)^1/2.

Remember when using the disc method the disks/washers are perpendicular to the axis of revolution.

When using the shell method the shells are parallel to the axis of revolution.

Rather that going into all the details here, instead take a look at this Desmos graph, enabling and expanding each folder one at a time.

desmos.com/calculator/muqhazlqht

This question was posed over 12 years ago, but perhaps someone will stumble upon this answer and be able to make use of the graph!