AP Calculus BC
4.2 Area
Approximating the area of a region in the plane
Left Riemann Sum
Right Riemann Sum
Midpoint Riemann Sum
1) Approximate the left Riemann sum for the area of the region bounded by the graph
of f(x) = x2, the x-axis, and the vertical lines x = 0 and x = 2, using 4 rectangles.
2) Approximate the right Riemann sum for the area of the region bounded by the graph
of f(x) = x  2 , the x-axis, and the vertical lines x = 0 and x = 2, using 8 rectangles.
3) Approximate the midpoint Riemann sum for the area of the region bounded by the
graph of f(x) = tan x, the x-axis, and the vertical lines x = 0 and x =

, using 4
4
rectangles.
4.3 Riemann Sums and Definite Integrals
Partitions do not have to be the same width.
Example 1): Let f(x) = 8 + 12 sinx – 4x.
Find the Riemann sum over the closed interval [0, 4] if the partition is x0 = 0 < x1 = 1<
x2 =1.8 < x3 =2.9 < x4 = 4 and intermediate points x = {0.4, 1.2, 2, 3.5}
Riemann Sum
If f is continuous and non-negative on the interval [a, b], then the area of the region
bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is
b
Area =  f ( x )dx
a
Indefinite Integral
 f ( x )dx  F ( x )  C (Family of functions)
Definite Integral
b
 f ( x )dx  Numerical Value
a
Continuity Implies Integrability
If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].
Two ways to evaluate a definite integral, at this time –
1) rectangle methods
2) use a geometric formula
A third way to evaluate definite integrals will be introduced in Section 4.4.
Sketch the region corresponding to each definite integral. Then evaluate each integral
using a geometric formula.
3
2)  4dx
3
3)  ( x  2)dx
0
1
2
2
4)
2
 ( 4  x )dx
2
5)
 x  1 dx
0
Properties of Definite Integrals
a
A) If f is defined at x = a, then  f ( x )dx  0
a
1
 xdx 
1
a
b
b
a
B) If f is integrable on [a, b], then  f ( x )dx    f ( x )dx
0
 ( x  2)dx 
3
C) If f is integrable on the three closed intervals determined by a, b, and c, then
b
c
b
a
a
c
 f ( x )dx   f ( x )dx   f ( x )dx
4
 x  3 dx
=
0
b
D) If f is integrable and nonnegative on the closed interval [a, b], then  f ( x )dx  0 .
a
3
 ( x  2)dx
0
E) If f and g are integrable on the closed interval [a, b] and f ( x )  g ( x ) for every x in
b
b
a
a
[a, b], then  f ( x )dx   g ( x )dx .
3
6) Evaluate  (  x 2  4 x  3)dx using each of the following values.
1
3
26
 x dx  3
1
2
3
6
0
3
3
3
1
1
 xdx  4
 dx  2
7) Given  f ( x )dx  4 and  f ( x )dx  1 evaluate:
6
a)  f ( x )dx
3
b)  f ( x )dx
0
6
3
6
c)  f ( x )dx
3
d)  5f ( x )dx
3
12
8) Approximate the value of
 f ( x )dx
with 5 subintervals as indicated in the chart.
0
Assume the function is an increasing function.
x
0
3
5
8
10
12
f(x)
0
4
7
10
14
17
a) Left Riemann sum:
b) Right Riemann sum:
9) The graph of f consists of line segments, as shown in the figure. Evaluate each
definite integral by using geometric formulas.
1
a)  f ( x )dx
d)
4
b)  3f ( x )dx
7
c)  f ( x )dx
0
3
0
11
11
10
 f ( x )dx
5
e)
 f ( x )dx
0
f)
 f ( x )dx
4