Deck 6: Integration

Question 1

Write sigma notation of 4 - 9 + 16 - 25 +... + .

A)
B)
C)
D)
E)

Answer 1

Question 2

Evaluate the sum .

A) 420
B) 70
C) 67
D) 417
E) 356

Answer 2

420

Question 3

Evaluate .

A) 1 +
B)
C) -
D) 1 -
E) -

Answer 3

-

Question 4

Evaluate the .

A) +
B) -
C)
D) 2 -
E) 2 +

Question 5

Find and evaluate the sum .

A)
B) -
C)
D) -
E)

Question 6

Evaluate .

A) -1
B) 0
C) 51
D) 1
E) 101

Question 7

Express the sum + + + +..... + using sigma notation.

A)
B)
C)
D)
E)

Question 8

Simplify the expression .

A) ln((2n)!)
B)
C) (2 ln n)!
D) 2 ln(n!)
E) (ln(n))!

Question 9

Express the sum in the series .

A) 2 + 9 + 7k
B) 2 + 9 + 5k
C) 3 + 9 + 7k
D) 3 + 9 + 5k
E) 2 - 9 + 7k

Question 10

Evaluate the sum Hint: = - .

A) 1
B)
C)
D)
E)

Question 11

Evaluate the sum .

A)
B)
C)
D)
E)

Question 12

Express the sum as a polynomial function of n.

A) 3 + + n
B) 3 + - n
C) 3 + + 4n
D) 3 + 3 - n
E) 3 - - n

Question 13

Find an approximation for the area under the curve y = 1 - and above the x-axis fromx = 0 to x = 1 using a sum of areas of four rectangles each having width 1/4 and (a) tops lying under the curve, or (b) tops lying above the curve. What does this tell you about the actual area under the curve?

A) (a) , (b) ; < area under curve <
B) (a) , (b) ; < area under curve <
C) (a) , (b) ; < area under curve <
D) (a) , (b) ; < area under curve <
E) (a) , (b) ; < area under curve <

Question 14

Given that the area under the curve y = and above the x-axis from x = 0 to x = a > 0 is square units, find the area under the same curve from x = -2 to x = 3.

A) square units
B) square units
C) 9 square units
D) 6 square units
E)

Question 15

Construct and simplify a sum approximating the area above the x-axis and under the curve y = between x = 0 and x = 3 by using n rectangles having equal widths and tops lying under or on the curve. Find the actual area as a suitable limit.

A) , area = 9 square units
B) , area = 9 square units
C) , area = 6 square units
D) , area = 6 square units
E) , area = 9 square units

Question 16

Construct and simplify a sum approximating the area above the x-axis and under the curvey = between x = 0 and x = 3 by using n rectangles having equal widths and tops lying above or on the curve. Find the actual area as a suitable limit.

A) , area = 9 square units
B) , area = 9 square units
C) , area = 6 square units
D) , area = 6 square units
E) , area = 9 square units

Question 17

Write the area under the curve y = cos x and above the interval [0, $\pi$ /2] on the x-axis as the limit of a sum of areas of n rectangles of equal widths. Have the upper-right corners of the rectangles lie on the curve.

A) Area =
B) Area =
C) Area =
D) Area =
E) Area =

Question 18

Given that = , find the area under y = and above the interval [0, a] on the x-axis (where a > 0 ) by interpreting the area as a limit of a suitable sum.

A) square units
B) square units
C) square units
D) square units
E) square units

Question 19

The limit represents the area of a certain region in the xy-plane. Describe the region.

A) region under y = cos x, above y = 0, between x = 0 and x =
B) region under y = sin x, above y = 0, between x = 0 and x =
C) region under y = cos x, above y = 0, between x = 0 and x = $\pi$
D) region under y = sin x, above y = 0, between x = 0 and x = $\pi$
E) region under y = cos x, above y = 0, between x= and x = $\pi$

Question 20

By interpreting it as the area of a region in the xy-plane, evaluate the limit .

A) 2 + 2 $\pi$ (the area of the trapezoidal region under y = 1 + $\pi$ x, above y = 0 from x = 0 to x = 2)
B) 1 + $\pi$ (the area of the trapezoidal region under y = 1 + 2 $\pi$ x, above y = 0 from x = 0 to x = 1)
C) 2 + 4 $\pi$ (the area of the trapezoidal region under y = 1 + 2 $\pi$ x, above y = 0 from x = 0 to x = 2)
D) 4 + 2 $\pi$ (the area of the trapezoidal region under y = 2 + $\pi$ x, above y = 0 from x = 0 to x = 2)
E) 2 + (the area of the trapezoidal region under y = 2 + $\pi$ x, above y = 0 from x = 0 to x = 1)

Question 21

By interpreting it as the area of a region in the xy-plane, evaluate the limit .

A) $\pi$ (the area of a quarter of a circular disk of radius 2)
B) 2 $\pi$ (the area of half of a circular disk of radius 2)
C) 4 $\pi$ (the area of a circular disk of radius 2)
D) 8 $\pi$ (the area of half of a circular disk of radius 4)
E) 16 $\pi$ (the area of a circular disk of radius 4)

Question 22

Let P denote the partition of the interval [1, 3] into 4 subintervals of equal length x = 1/2.Evaluate the upper and lower Riemann sums U(f,P) and L(f,P) for the function f(x) = 4x².

A) U(f,P) = 40, L(f,P) = 30
B) U(f,P) = 41, L(f,P) = 29
C) U(f,P) = 42, L(f,P) = 28
D) U(f,P) = 43, L(f,P) = 27
E) U(f,P) = 44, L(f,P) = 26

Question 23

Let P denote the partition of the interval [1, 2] into 8 subintervals of equal length x = 1/8.Evaluate the upper and lower Riemann sums U(f P) and L(f,P) for the function f(x) = 1/x.Round your answers to 4 decimal places.

A) U(f,P) = 0.7110, L(f,P) = 0.6781
B) U(f,P) = 0.7254, L(f,P) = 0.6629
C) U(f,P) = 0.7302, L(f,P) = 0.6571
D) U(f,P) = 0.7378, L(f,P) = 0.6510
E) U(f,P) = 0.7219, L(f,P) = 0.6683

Question 24

Let P denote the partition of the interval [1, 4] into 6 subintervals of equal length x = 1/2.Evaluate the upper and lower Riemann sums U(f, P) and L(f,P) for the function f(x) = .Round your answers to 4 decimal places.

A) U(f,P) = 4.9115, L(f,P) = 4.4115
B) U(f,P) = 4.9135, L(f,P) = 4.4109
C) U(f,P) = 4.9180, L(f,P) = 4.4057
D) U(f,P) = 4.9002, L(f,P) = 4.4250
E) U(f,P) = 4.9183, L(f,P) = 4.4093

Question 25

Calculate the upper Riemann sum for f(x) = + 1 corresponding to a partition P of the interval [0, 3] into n equal subintervals of length 3/n. Express the sum in closed form and use it to calculate the area under the graph of f, above the x-axis, from x = 0 to x = 3.

A) U(f,P) = + 3, area = 12 square units
B) U(f,P) = + 3, area = 12 square units
C) U(f,P) = + 3, area = 12 square units
D) U(f,P) = + 3, area = 12 square units
E) U(f,P) = + 3, area = 12 square units

Question 26

Calculate the lower Riemann sum for f(x) = corresponding to a partition P of the interval [0, 1] into n equal subintervals of length 1/n. Given that n = 1 (which can be verified by using l'Hopital's Rule), find the area under y = and above the x-axis between x = 0 and x = 1.

A) L(f,P) = , area = e square units
B) L(f,P) = , area = square units
C) L(f,P) = , area = e - 1 square units
D) L(f,P) = , area = square units
E) L(f,P) = , area = square units

Question 27

Express dx as a limit of Riemann sums corresponding to partitions of [0, 1] into equal subintervals and using the values of f at the midpoints of the subintervals.

A) dx =
B) dx =
C) dx =
D) dx =
E) dx =

Question 28

Express dx as a limit of lower Riemann sums corresponding to partitions of[0, 2] into equal subintervals.

A) dx = .
B) dx = .
C) dx = .
D) dx = .
E) dx = .

Question 29

Write the following limit as a definite integral: .

A)
B)
C)
D)
E)

Question 30

Write the following limit as a definite integral:

A)
B)
C)
D)
E)

Question 31

Use the limit definition of definite integral to evaluate dx.

A) 0
B) -
C)
D) 1
E) -1

Question 32

Use the limit definition of the definite integral to evaluate dx.

A) 10
B) 18
C) 6
D) 30
E) 9

Question 33

Write the following limit as a definite integral:

A) dx
B) dx
C) dx
D) dx
E) dx

Question 34

Given that and = -1, find

A) -3
B) -1
C) 3
D) 1
E) -2

Question 35

Suppose that

A) -5
B) -3
C) -7
D) -1
E) 7

Question 36

Evaluate dx.

A) 42
B) 0
C) 21
D) 51
E) 16

Question 37

If f and g are integrable functions on the interval [a, b], then = . dx.

Question 38

Question 39

If f(x) is an even function and g(x) is an odd function ,both of which are integrable over the interval [-a, a], then

Question 40

Evaluate (2 - ) dx by interpreting the integral as representing an area.

A)
B) 4
C) 2
D)
E) -

Question 41

Evaluate dx by interpreting the integral as representing an area.

A) 8 $\pi$
B) 4 $\pi$
C) 16 $\pi$
D) 8
E) 16

Question 42

Given that dx = , evaluate dx.

A) $\pi$ /4
B) $\pi$ /2
C) $\pi$
D) 1/2
E)

Question 43

Given the piecewise continuous function f(x) = evaluate by using the properties of definite integrals and interpreting integrals as areas.

Question 44

If f(x) is an even function integrable on the closed interval [0 , 2a] , a > 0 , then =2 .

Question 45

Find the average value of the function f(x) = sin (x/2) + $\pi$ on [- $\pi$ , $\pi$ ].

A) $\pi$
B)
C) 2 $\pi$
D)
E) 2

Question 46

The velocity of a particle moving along a straight line at time t is given by v(t) = t² - 6t + 8 m/s.Find the distance travelled by the particle from t = 0 to t = 3.

A) m
B) 21 m
C) 6 m
D) 4 m
E) 33 m

Question 47

What values of a and b, satisfying a < b, maximize the value of

A) a = 0, b = 1
B) a = -1, b = 1
C) a = 0, b = 2
D) a = - $\infty$ , b = $\infty$
E) a = -1, b = 0

Question 48

What values of a and b, satisfying a < b, maximize the value of

A) a = -2, b = 4
B) a = 0, b = 4
C) a = - $\infty$ , b = $\infty$
D) a = -2, b = 0
E) a = 1, b = 3

Question 49

Evaluate the definite integral

A) -
B)
C)
D) -
E) -12

Question 50

Compute the definite integral - 2x + 1)dx.

A) 14
B) 22
C) 21
D) 24
E) 20

Question 51

Compute the integral - x)dx.

A) 4 -
B) 4 +
C) 4 -
D) 4 +
E) 2 -

Question 52

Compute the integral

A)
B)
C) -
D) -
E)

Question 53

Evaluate the integral dx.

A)
B) 1
C)
D)
E)

Question 54

Find dx.

A)
B)
C)
D)
E)

Question 55

Evaluate the integral

A)
B)
C)
D)
E) -

Question 56

Find the average value of the function f(x) = + 3 - 2 - 3x + 1 on the interval [0, 2].

A)
B)
C)
D)
E)

Question 57

Find the average value of the function f(x) = sin x on [0, 3 $\pi$ /2].

A)
B)
C)
D)
E)

Question 58

Evaluate the definite integral dx.

A) 11.1
B) 9.9
C) 10.1
D) 15
E) -10.1

Question 59

Evaluate the integral dx.

A) 4
B) 6
C) 7
D) 5
E) 3

Question 60

dx = 4

Question 61

Find the average value of the function f(x) = 3x, over the interval [- $\pi$ /12, $\pi$ /12].

A)
B)
C)
D)
E)

Question 62

Let f(t) = Evaluate dt.

Question 63

Evaluate the definite integral dx.

A) -
B)
C)
D) -33
E)

Question 64

Find the derivative of F(x) =

A) 2 ln x
B) 0
C) ln(u)
D) ln x
E) ln x

Question 65

Find the derivative of F(x) = dt.

A) 2 cos ( )
B) 2 sin ( )
C) 2 cos ( )
D) cos ( )
E) cos ( )

Question 66

Given that the relation 3 + dt = 3 defines y implicitly as a differentiable function of x, find .

A)
B)
C) 6x + cos (t) - t sin (t)
D) 6x +cos(y) - ysin (y)
E) 6x + ycos (y)

Question 67

Evaluate

A) -
B) -
C)
D) 1
E)

Question 68

Find the point on the graph of the function f(x) = dt where the graph has a horizontal tangent line.

A)
B)
C)
D) (1, 0)
E)

Question 69

: is equal to

A) 2x
B) -
C) 2x
D) 2x
E) 2x

Question 70

Evaluate the integral dx.

A) + C
B) + C
C) + C
D) - + C
E) - + C

Question 71

Find the inflection point of the function f(x) = dt, where x > 0.

A) (1, 0)
B) (e, )
C) (e, 1)
D) (e, )
E)

Question 72

Evaluate the integral dx.

A) ln(3 ) + C
B) ln + C
C) 3ln + C
D) + C
E) + C

Question 73

Evaluate the integral dx.

A) + C
B) + C
C) - + C
D) - + C
E) + C

Question 74

Evaluate the integral

A) -
B) +
C) -
D) +
E)

Question 75

Evaluate the integral dx.

A) + C
B) + C
C) + C
D) + C
E) + C

Question 76

Evaluate the integral dx.

A) ln( + 4x + 5) + C
B) 2 ln( + 4x + 5) + C
C) ln( + 4x + 5) + C
D) -2 ln( + 4x + 5) + C
E) - ln( + 4x + 5) + C

Question 77

Evaluate the integral dx.

A) + C
B) + C
C) + C
D) + C
E) + C

Question 78

Evaluate dx.

Question 79

Evaluate the integral dx.

A) + C
B) -3 + C
C) + C
D) - + C
E) - + C

Question 80

Evaluate the integral cos ( ) dx.

A) + C
B) + C
C) - + C
D) - + C
E) + C