The area between the curves y = f(x) and y = g(x) and
between x = a and x = b is
An analogous formula holds for the area between the curves
y = c and y = d and x = f(y) and x = g(y):
For a solid S lying between x = a and x = b with
cross-sectional area function A(x) (A continuous),
the volume of S is
Again, an analogous formula holds for the volume of a
solid S lying between y = c and y = d with
cross-sectional area function A(y) (A continuous). In this case,
1.
2.
3. (6.1.50) Find the number a such that the line x = a
bisects the area under the curve y = 1/(x^2)
for 1≤x≤4. Also, find the number b such that the line y = b bisects the same
area.
Solution : We want our line x = a to split our areas. That
is, we want an a so that 
Taking the integrals, we find 
Solving for a ,
we find that 
For the second part, we want to split the areas using a
horizontal line y = b. A horizontal line
will give us a wedge above the line and a region below that includes a rectangle
sitting below the
line y = 1/16. The area of the lower region is 
The
first term comes
from the rectangle of length 3 between 1 and 4 and height 1/16. The second terms
comes from
determining the area between the curve 
and x =
1. The upper region has area given by
dy. We set these two equal to each other and
integrate to get
We simplify this to get 
To solve for b , we set c^2 = b and consider 
We solve for c using the quadratic formula to get
Thus
Squaring this we get
Solution: The solid looks like the space between a
funnel and a cylinder. We split this solid
into two different regions : the lower one consists of a cylinder with a hole and
the upper region
is a triangle rotated around the line x = 1. The lower region has
cross-sectional area
and the
upper region has cross-sectional area 
Then we find
5. Suppose 2≤f(x)≤4 for all x. Which of the following is
false:
(B) The antiderivative of f is always positive .
(D) The antiderivative of f is always negative .
Solution: (A) is true since f(x)≥2 for all x. (C) is true
since 2≤f(x)≤4 implies that
(E) is true since
For (B) and (D), note
that an antiderivative of f is
by the
Fundamental Theorem of Calculus. Since
f(x) > 0, g(x) > 0 for any x. Thus (B) is true while (D) is false. Hence the
answer is (E).
