Name______________________________________
Riemann Sums – an estimation of the integral.
b
Area =

a
n
f  x dx   f (ci )xi , where f (ci ) is the height of the rectangle and xi is the width of the
i 1
rectangle.
Left Riemann Sum – Use left endpoint of the rectangle as the height
Right Riemann Sum – Use the Right endpoint of the rectangle as the height
Midpoint Riemann Sum – Use the midpoint of the rectangle as the height
Trapezoidal Sum = Use trapezoids to find the area underneath the curve
5
Example 1: Using three rectangles estimate
 (3x  4)dx using
2
a) Left Riemann Sum
b) Right Riemann Sum
c) Midpoint Riemann Sum
d) Trapezoidal Sum
Example 2: Given
x
0
f(x)
2.6
20
3.9
35
4.6
45
4.3
50
3.8
50
a) Estimate
 f  x dx using a Left Riemann Sum with 4 intervals.
0
50
b) Estimate
 f  x dx using a Right Riemann Sum with 4 intervals.
0
50
c) Estimate
 f  x dx using a Midpoint Riemann Sum with 2 intervals.
0
50
d) Estimate
 f  x dx using a Trapezoidal Sum with 4 intervals.
0
Note: Can NOT use Riemann Program on your calculator if you have uneven intervals.
6
Example 3: Use the table to find the upper and lower estimate of
 f  x dx
0
x
f(x)
0
-6
2
8
4
30
6
80
Riemann Sums assignment
1. Approximate the area under the curve y 
a.
b.
c.
d.
1
over the interval [1,2]
x
Using a Left Riemann Sum with 5 rectangles (0.746)
Using a Right Riemann Sum with 5 rectangles (0.646)
Using a Midpoint Riemann Sum with 5 rectangles (0.692)
Find the Trapezoidal sum with 5 rectangles (0.696)

2. Estimate  sin xdx using the trapezoidal sum using four subintervals. (1.896)
0
x2
3. Use the trapezoidal sum using 5 subintervals to estimate the area under y  8 
over the
2
interval [0,6]
4.
Approximate the area under the curve y  x over the interval [0,1]
a. Using a Left Riemann Sum with 4 rectangles (0.518)
b. Using a Right Riemann Sum with 4 rectangles (0.768)
5. Use the left Riemann sum using 4 subintervals to estimate the area under y 
3
 1 over the
x2
interval [1,2] (0.808)
6. Use the Midpoint Riemann sum using 3 subintervals to estimate the area under y 
1
1
over the
x2

interval  , 2  (1.372)
2 
7. Use the Left Riemann Sum to approximate the area under the curve y   2 x  1 over the
2
interval [0,1] using 6 rectangles. (0.352)
8. Use the trapezoidal sum to approximate the area under the curve y  x3 over the interval [-2,4] using 3
rectangles. (72)
9. A table of f(x) is given below.
x
0
4
8
f(x)
7.5
9.0
9.3
12
9.5
16
8.8
20
8.0
24
7.2
24
a) Estimate
 f  x dx using a Left Riemann Sum with 6 intervals. (208.4)
0
24
b) Estimate
 f  x dx using a Right Riemann Sum with 6 intervals. (207.2)
0
24
c) Estimate
 f  x dx using a Midpoint Riemann Sum with 3 intervals. (212)
0
24
d) Estimate
 f  x dx using a Trapezoidal Sum with 3 intervals. (203.6)
0
10. An experiment was preformed in which oxygen was produced at a continuous rate. The rate at
which oxygen was produced was measured each minute and the results tabulated.
minutes
0
1
2
3
4
5
6
Oxygen
0
1.4
1.8
2.2
3.0
4.2
3.6
Use the trapezoid sum to estimate the total amount of oxygen produced in 6 minutes.
When n=6
14.4
When n=63 13.2
11. A table of f(x) is given below.
x
0
40
70
f(x)
150
180
195
90
184
100
172
100
a) Estimate
 f  x dx using a Left Riemann Sum with 4 intervals.
(17,140)
0
100
b) Estimate
 f  x dx using a Right Riemann Sum with 4 intervals.
(18,450)
0
100
c) Estimate
 f  x dx using a Midpoint Riemann Sum with 2 intervals. (18,120)
0
100
d) Estimate
 f  x dx using a Trapezoidal Sum with 4 intervals.
(17,795)
0
12. A table of f(x) is given below.
x
0
20
40
f(x)
1.2
2.8
4.0
60
4.7
80
5.1
100
5.2
120
4.8
120
a) Estimate
 f  x dx using a Left Riemann Sum with 6 intervals.
(460)
0
120
b) Estimate
 f  x dx using a Right Riemann Sum with 6 intervals. (532)
0
120
c) Estimate
 f  x dx using a Midpoint Riemann Sum with 3 intervals. (508)
0
120
d) Estimate
 f  x dx using a Trapezoidal Sum with 3 intervals
(484)
0
13. The graph of the function f over the interval [1.7] is shown. Using values from te graph find
7
the trapezoidal sum estimate for the integral
 f  x dx by using the indicated number of
1
subintervals.
a. n=3 (25)
b. n=6 (24.5)
14. The graph of f over the interval [1,9] is shown in the figure. Using the data in the figure find
9
the midpoint approximation with 4 equal subdivisions for
 f  x dx . (24)
1