Expert: Paul Klarreich - 12/28/2007

Question
The link to the graph is:
http://img409.imageshack.us/img409/2923/areaoi2.jpg

The shaded regions R1 and R2 shown above are enclosed by the graphs of f(x) = x^2 and g(x) = 2^x.

a. Find the x- and y-coordinates of the three points of intersection of the graphs of f and g.
b. Without using absolute value, set up an expression involving one or more integrals that gives the
total area enclosed by the graphs of f and g. Do not evaluate.
c. Without using absolute value, set up an expression involving one or more integrals that gives the
volume of the solid generated by revolving the region R1
about the line y = 5. Do not evaluate.

Answer
Questioner:   Stephano
Category:  Calculus
Private:  No
 
Subject:  AP Calculus
Question:  The link to the graph is:
http://img409.imageshack.us/img409/2923/areaoi2.jpg

The shaded regions R1 and R2 shown above are enclosed by the graphs of f(x) = x^2 and g(x) = 2^x.

a. Find the x- and y-coordinates of the three points of intersection of the graphs of f and g.
b. Without using absolute value, set up an expression involving one or more integrals that gives the total area enclosed by the graphs of f and g. Do not evaluate.
c. Without using absolute value, set up an expression involving one or more integrals that gives the
volume of the solid generated by revolving the region R1
about the line y = 5. Do not evaluate

...........................................
Hi, Stephano,

I got the picture. In it, the green graph is g = 2^x and the blue is f =x^2.

The green (g) is higher from -0.766 to 2 and the blue is higher from 2 to 4.

So you will integrate:

{2
|    g - f PLUS
}-0.766

{4
|    f - g
}2

for the area.

Your picture does not make clear which is R1.  I will assume it is the part from -0.766 to 2.

If you revolve that about y = 5, your 'typical' volume element is a DISK-like ring.

The inner radius, r, is  5 - f(x).
The outer radius, R, is  5 - g(x).
The area is  pi(R^2 - r^2)
The thickness is dx.

The volume will be:

{2
|   pi[ (5 - f(x))^2 - (5 - g(x))^2) dx
}-0.766

That should do it. (Put in your f and g.)