Problem
23: 1998-1999
Find the volume of the solid whose base is bounded by
the graphs of y = x + 1 and y = x2 
1, with square cross sections taken perpendicular to the x-axis.
Difficulty rating: 
Problem and image from Calculus by Larson, Hostetler
and Edwards (c) 1994
Solution:
In the below picture, I have drawn the base of the figure, looking
down from above the shape towards the x-y plane. The green rectangle represents
the bottom-most side of the red square in the above picture.
Thus, the square will have all its sides the length of the
green rectangle, which is found by subtracting the lower curve from the
upper curve to get 
. Don't
forget that to find a volume with a known cross-section, we integrate the
area of one cross-section, in this case with x-boundaries since the green
rectangle is vertical. The area of a square cross-section is (side)2,
so we get:
.
The x-boundaries come from setting the two equations equal
to one another (to find intersection points). The final solution will be
81/10 or 8.1.
