Question : 31) f(x) = x^2 - 6x + 9; [2, 4] : 2151784

Use the definite integral to find the area between the x-axis and f(x) over the indicated interval.

31) f(x) = x² - 6x + 9; [2, 4]

A) (1/3)

B) (2/3)

C) (7/3)

D) (4/3)

32) f(x) = 2x - x²; [0, 2]

A) (5/3)

B) (4/3)

C) (7/3)

D) (2/3)

33) f(x) = (3/x³); [1, 3]

A) (1/3)

B) 3

C) (4/3)

D) (1/2)

34) f(x) = -x² + 9; [0, 5]

A) (10/3)

B) (5/9)

C) (98/3)

D) (10/9)

35) f(x) = x² + 1; [0, 1]

A) (2/3)

B) (4/3)

C) (5/3)

D) (1/3)

36) f(x) = x⁴ - 4x³ + 4x²; [0, 2]

A) (17/15)

B) (16/15)

C) (15/17)

D) (15/16)

Find the area of the shaded region.

37) y = 2x + 1

A) 7.5

B) 10

C) 5

D) 12.5

38) y = x² + 3

A) (26/3)

B) (23/3)

C) (25/3)

D) (22/3)

39)

y = 4 - x²

A) 5

B) 3

C) (23/3)

D) (5/3)

40)

A) (38/3)

B) (22/3)

C) (16/3)

D) (29/3)

41) y = x³ - 4x

A) (17/4)

B) (33/4)

C) (41/4)

D) (9/4)

42) y = (1/x)

A) ln5.5

B) ln4.5

C) ln5

D) ln4

43) y = (x - 3)²

A) (1/3)

B) (5/3)

C) (2/3)

D) (4/3)

44) y = e^x

A) e² + e - 1

B) e² + e

C) e² - e

D) e² - e + 1

45)

A) 14

B) 7

C) 49

D) 42

Solve the problem.

46) A company has found that its rate of expenditure (in hundreds of dollars) on a certain type of job is given by

E'(x) = 6x + 12,

where x is the number of days since the start of the job. Find the total expenditure if the job takes 6 days.

A) $4800

B) $180

C) $18,000

D) $48

47) After a new firm starts in business, it finds that its rate of profit (in hundreds of dollars) after t years of operation is given by

P'(t) = 3t² + 12t + 10.

Find the profit in year 4 of the operation.

A) $15,200

B) $8900

C) $15,550

D) $11,100

48) A certain object moves in such a way that its velocity (in m/s) after time t (in s) is given by

v = t² + 5t + 2.

Find the distance traveled during the first four seconds by evaluating Round your answer to the nearest tenth of a meter.

A) 48.0 m

B) 61.3 m

C) 38.0 m

D) 69.3 m

49) For a particular circuit, the current (in amperes) after time t (in seconds) at a certain point P is given by

i = 0.005t^0.22.

Find the charge (in coulombs) that passes point P during the first second by evaluating A) 0.005 coulombs

B) 0.0041 coulombs

C) 1.22 coulombs

D) 244 coulombs

50) A force acts on a certain object in such a way that when the object has moved a distance of r (in m), the force f (in newtons) is given by

f = 7r² + 5r.

Find the work (in joules) done through the first four meters by evaluating A) 189.3 joules

B) 154.3 joules

C) 40 joules

D) 64 joules

51) An object is traveling with a velocity (in feet per second) given by

v(t) = 8t³ - 3t² + 2t,

where t is time in seconds. Find the object's average velocity from t = 0 to t = 8 seconds.

A) 490.0 ft/sec

B) 1285.3 ft/sec

C) 7744.0 ft/sec

D) 968.0 ft/sec

52) A population of bacteria grows at a rate of P'(t) = 14 e^t where t is time in hours. Determine how much the population increases from t = 0 to t = 3. Round your answer to two decimal places.

A) 548.4

B) 274.20

C) 267.2

D) 281.2

53) The rate of change in a person's body temperature, with respect to the dosage of x milligrams of a drug, is given by D'(x) = (7/x + 6). One milligram raises the temperature 2.3°C. Find the function giving the total temperature change.

A) D(x) = 7ln|x + 6| - 11.3

B) D(x) = 7ln| x + 6| + 2.3

C) D(x) = ln|(7/x + 6)| - 2.3

D) D(x) = ln|(7/x + 6)| - 11.3

54) The number of books in a small library increases at a rate according to the function B'(t) = 178e^0.02t, where t is measured in years after the library opens. How many books will the library have 8 year(s) after opening?

A) 1544

B) 31

C) 209

D) 8900

55) For a certain drug, the rate of reaction in appropriate units is given by R'(t) = (6/t) + (6/t²), where t is measured in hours after the drug is administered. Find the total reaction to the drug from t = 3 to t = 9. Round to two decimal places, if necessary.

A) 7.93

B) 8.56

C) 18.78

D) 17.11

Provide the proper response.

56) If f(x) ≥ 0 on the interval [a, b], then represents