Section 7.6 – Numerical Integration

Day 5:

I can integrate definite integrals using Left Hand Sum, Right Hand Sum, Midpoint Sum, and Trapezoidal Rule.

Fill in the blank (try to do it using your memory):

1.

2.

3.

8

4

3dx

xrepresents 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

3dx

x

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

3dx

x

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

3dx

x

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

3dx

x

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 0.1

A 1 2.0912

5 0.42857) 0.375

Approximate using left-hand sums of four rectangles of equal width

22

x

0

e dx

x f(x)0 1

0.5 1.2841 2.7183

1.5 9.4877

1 1.284 2.7181

A 7.3 9.4877 2452

Approximate using trapezoidal rule with n = 4 2

0

sin x dx

x f(x)0 0

0.195090.707110.98079

0

/ 4/ 2

3 / 4

0 2 0.19509 0.70711 0.9801

A 0.834382

9 04

7

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 38

2

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 18518.46

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 1F

2 2

Consider the function f whose graph is shown below. Use theTrapezoid Rule with n = 4 to estimate the value of

9

1

f x dx

A. 21 B. 22 C. 23 D. 24 E. 25

13 2 1 4 2 5 22

22 B

X

X

X

X

X

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. 2.5 0 T

III. True

CALCULATOR REQUIRED

53

1

When x x 1 is approximated by using the mid-points

of three rectangles of equal width, the approximation is nearest to

A. 22.6 B. 22.9 C. 23.2 D. 23.5 E. 23.8

x 0 2 4

f(x) 1 2.646 7.810

2 1 2.646 7.810 22.912

The graph of f over the interval [1, 9] is shown in the figure.Find a midpoint approximation with four equal subdivisions for

9

1

f x dx

A. 20 B. 21 C. 22 D. 23 E. 24

X

XX X

2 2 4 3 3 24

CALCULATOR REQUIRED

Let 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 4

f(x) 0 0.693 1.099 1.386

11 0 2 0.693 1.099 1.386

2

Trapezoidal Rule:

1altitude sum of bases

2 1 2 3 n

1x y 2 y y ... y

2

Error in Trapezoidal Rule:

3b

22

a

2

M b af x dx Trap n

n

where M is the maximum value of

12

f" x

Midpoint Rule midpt. altitude sum of bases

Error in Midpoint Rule:

3b

22

a

2

M b af x dx Mid n

n

where 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

3

22

b aM

24n 23 4

6 18f ' x f " x f " 1 18 M

x x 34 1 27

2

18 270.001

24n

223142.85 n

151.13 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

3

22

M b a

12n 22 3

5 10f ' x f " x f " 1 10 M

x x

33 1 8

2

10 80.01

12n2n 666.67

n 25.820

26