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