TA: Hamza Ruzayqat
Math 142, Calculus II, Fall 2016
1
Find the area of the wedge in the following figure by integrating the area of vertical cross sections.
c
0
h
b
x
`
a
Solution:
We need to find the area of the vertical cross section, i.e., the area of the small blue triangle.
Let’s denote this are by A(x), where x goes from 0 to a. Then,
A(x) =
1
1
(base) · (height) = ` h.
2
2
Notice that the small triangle that has base of length ` and height a − x is similar to the big
green triangle, the one that has base of length a and a height b. Thus,
b
`
=
a−x
a
a−
x
=⇒
b
` = (a − x).
a
b
x
a
Moreover, the small triangle that has a base of length a − x and a height h is similar to the
big pink triangle that has base of length a and height c. Thus,
c
h
c
c
=
=⇒ h = (a − x).
a−x
a
a
So we got h and ` and therefore,
a a−
x
x
0
A(x) =
1
1 b
c
bc
bc
`h =
(a−x) (a−x) = 2 (a−x)2 = 2 (a2 −2ax+x2 ).
2
2 a
a
2a
2a
TA: Hamza Ruzayqat
Math 142, Calculus II, Fall 2016
2
Integrating from x = 0 to x = a, we have
Z a
Z a
bc
bc h 2
1 3 ia
2
2
2
A(x) dx = 2
(a − 2ax + x ) dx = 2 a x − ax + x
V =
2a 0
2a
3
0
0
3
bc a
abc
= 2
=
.
2a 3
6
J