Math 152, Fall 2008
Lecture 2.
08/28/2008
Example 1. Evaluate each integral.
(a)
Z4
1
x2
r
1
1 + dx
x
(b)
e
1
(c)
Z
sin x
dx
1 + cos2 x
Zπ/2
(e)
x 2 sin x
dx
1 + x6
−π/2
(j)
Z
Ze 4
arcsin x + x
√
dx
1 − x2
(d)
(f)
R
Z
dx
√
x ln x
t 2 cos(1 − t 3 )dt
x2
dx
+ a2
Chapter 7. Applications of integration
Section 7.1 Areas between curves
Consider the region S that lies between two curves y = f (x) and
y = g (x) and between the vertical lines x = a and x = b, where f
and g are continuous functions and f (x) ≥ g (x) for all x in [a, b].
y
y = f (x)
S
0
a
b
x
y = g(x)
S = {(x, y ) : a ≤ x ≤ b, g (x) ≤ y ≤ f (x)}
Let P be a partition of [a, b] by points xi and choose points xi∗ in
[xi −1 , xi ]. Let ∆xi = xi − xi −1 and kPk = max{∆xi }.
n
P
The Riemann sum
[f (xi∗ ) − g (xi∗ )]∆xi is an approximation to
i =1
what we intuitively think as the area of S.
This approximation appears to become better and better as
kPk → 0. We define the area A of S as
n
P
[f (xi∗ ) − g (xi∗ )]∆xi Therefore:
A = lim
kPk→0 i =1
The area of the region bounded by the curves y = f (x), y = g (x),
and the lines x = a and x = b, where f and g are continuous
functions and f (x) ≥ g (x) for all x in [a, b], is
Rb
A = [f (x) − g (x)]dx
a
Example 2. Find the area of the region bounded by the
parabolas y = 4x 2 , y = x 2 + 3
In general case, the area between the curves y = f (x), y = g (x)
Rb
and between x = a and x = b, is A = |f (x) − g (x)|dx
a
Example 3. Find the area of the region bounded by the curves
y = cos x, y = sin 2x an the lines x = 0 and x = π/2.
Example 4. Find the area of the shaded region.
y
y
=
x
x = y2 − 2
0
x
If a region is bounded by curves with equations x = f (y ),
x = g (y ), y = c and y = d, where f and g are continuous
functions and f (y ) ≥ g (y ) for all y in [c, d], then its area is
Rd
A = [f (y ) − g (y )]dy
c
y
d
S
x = f (y)
x = g(y)
c
0
x