Section 7.6 – Numerical Integration
8
4
3 dxx
represents the area between the curve 3/x and the x-axisfrom x = 4 to x = 8
Four Ways to Approximate the Area Under a CurveWith Riemann Sums
Left Hand SumRight Hand Sum
Midpoint SumTrapezoidal Rule
Approximate using left-hand sums of four rectangles of equal width
8
4
3 dxx
1. Enter equation into y12. 2nd Window (Tblset)3. Tblstart: 44. Tbl: 15. 2nd Graph (Table)
x f(x)4 0.755 0.66 0.57 0.42857
0.75 0.6 0.5 0.42857A 1 2.279
Approximate using right-hand sums of four rectangles of equal width
8
4
3 dxx
1. Enter equation into y12. 2nd Window (Tblset)3. Tblstart: 54. Tbl: 15. 2nd Graph (Table)
x f(x)5 0.66 0.57 0.428578 0.375
0.6 0.5 0.42857 0.375A 1 1.904
Approximate using midpoint sums of four rectangles of equal width
8
4
3 dxx
1. Enter equation into y12. 2nd Window (Tblset)3. Tblstart: 4.54. Tbl: 15. 2nd Graph (Table)
x f(x)4.5 0.666675.5 0.545456.5 0.461547.5 0.4
0.66667 0.54545 0.46154A 1 2.070.4 4
Approximate using trapezoidal rule with four equalsubintervals
8
4
3 dxx
1. Enter equation into y12. 2nd Window (Tblset)3. Tblstart: 44. Tbl: 15. 2nd Graph (Table)
4 0.755 0.66 0.57 0.428578 0.375
0.75 2(0.6 0.5 0.428571A 1 2.1040.3752
)
For the function g(x), g(0) = 3, g(1) = 4, g(2) = 1, g(3) = 8, g(4) = 5, g(5) = 7, g(6) = 2, g(7) = 4. Use the trapezoidal rule with n = 3 to estimate
7
1
g x dx
x g(x)1 43 85 77 4
1A 2 4 2 8 7 4 382
If the velocity of a car is estimated at 4 2v t t 3t 1
estimate the total distance traveled by the car from t = 4 to t = 10using the midpoint sum with four rectangles
104 2
4
t 3t 1dt t v(t)
4.75 442.386.25 1409.77.75 3428.39.25 7065.3
A 1.5 442.38 1409.7 3428.3 7065.3 18518.52
Consider the function f whose graph is shown below. Use theTrapezoid Rule with n = 4 to estimate the value of
9
1f x dx
A. 21 B. 22 C. 23 D. 24 E. 25
1 3 2 1 4 2 5 222
2 B
X
X
X
X
X
The graph of f is shown to the right. Which of the followingStatements are true?
2
0
0 3
1 2
I. f ' 3 f ' 1
II. f x dx f ' 3.5
III. f x dx f x dx
A. I only B. II only C. I and II only D. II and III only E. I, II, III
1 1 F
0 1 T
1 1 F2 2
A graph of the function f is shown to the right. Which of thestatements are true?
2
1
h 0
I. f 1 f ' 3
II. f x dx f ' 3.5
f 2 h f 2 f 2.5 f 2III. lim
h 2.5 2
A. I only B. II only C. I and II only D. II and III only E. I, II, III
I. 1 ? T
II. 6 0 T III. True
The graph of f over the interval [1, 9] is shown in the figure.Find a midpoint approximation with four equal subdivisions for
9
1f x dx
A. 20 B. 21 C. 22 D. 23 E. 24
X
XX X
2 2 4 3 3 24
CALCULATOR REQUIREDLet R be the region in the first quadrant enclosed by the x-axisand the graph of y = ln x from x = 1 to x = 4. If the Trapezoidrule with three subdivisions is used to approximate the area of R, the approximation is A. 1.242 B. 2.485 C. 4.970 D. 7.078 E. 14.156
X 1 2 3 4f(x) 0 0.693 1.099 1.386
1 1 0 2 0.693 1.099 1.3862
Trapezoidal Rule:
1 altitude sum of bases2
1 2 3 n1 x y 2 y y ... y2
Error in Trapezoidal Rule:
3b2
2a
2
M b af x dx Trap n
nwhere M is the maximum value of
12 f" x
Midpoint Rule midpt. altitude sum of bases
Error in Midpoint Rule:
3b2
2a
2
M b af x dx Mid n
nwhere M is the maximum value of
24 f" x
CALCULATOR REQUIRED
Determine how many subdivisions are required with the MidpointRule to approximate the integral below with error less than 0.001
4
21
3x
32
2
b aM24n
23 46 18f ' x f " x f " 1 18 M
x x 34 1 27
2
18 270.001
24n 223142.85 n151.13 n
152
CALCULATOR REQUIRED
Determine how many subdivisions are required with the TrapezoidRule to approximate the integral below with error less than 0.01
3
1
5x
32
2
M b a12n
22 35 10f ' x f " x f " 1 10 M
x x
33 1 8
2
10 80.01
12n2n 666.67
n 25.820
26