Topic Riemann Sum
Let’s demonstrate this concept with an example right out of the gate.
Example 1: Find the area under the graph of y = f (x), above the x-axis on the interval [2,10] given the
following data and using an right end sum approach. Let n = 4.
x
2
3
5
8
10
f (x)
5
7
11
10
6
Example 2: Using the Midpoint Rule to approximate the area under f ( x) =
the interval [2,6] using 4 rectangles of equal width.
8
and above the x-axis on
x +1
2
The Trapezoidal Rule
Let f be continuous on [a,b]. The Trapezoidal Rule for approximating the area under a
curve is
b-a
Area »
[ f (x0 ) + 2 f (x1 ) + 2 f (x2 ) + 2 f (x3 ) + 2 f (x4 ) + ! + 2 f (xn-1 ) + f (xn )]
2n
Example 3:
Using the Trapezoidal Rule to approximate the area under f ( x) = sin x
and above the x-axis
on the interval [0,π] using 4 trapezoids.
Example 4: The following table shows the speed in miles per hour of a cyclist at various times.
Time (min)
0 2 5 6 9 10 12
Speed (mph) 33 25 27 13 21 5 9
Use a trapezoidal approximation to find the distance (in miles) the cyclist traveled in the 12-minute time
interval.
Definite Integrals – Part I
Example 6: Evaluate the following definite integral:
1
ò 2x dx
-2
Example 7: Set up a definite integral that would calculate the area of each region below.
a.
b.
Example 8: Sketch and shade the region whose area is given by the definite integral. Use geometry to find
the area.
3
0
a.
ò (2 x + 5)dx
ò
b.
9 - x 2 dx
-3
-2
y
y
4
4
3
3
2
2
1
1
x
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
x
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
1
c.
ò (1 - x )dx
-1
2
y
1
x
-2
-1
1
2
-1
-2
Example 9: Given the graph of f(x). Find the solution to each of the following definite integrals.
0
a.
ò (- f ( x))dx
6
b.
-4
ò f ( x + 2)dx
-2
f ( x) dx
2
3
1
c.
ò
d.
ò [ f ( x) + 2] dx
0