MCQ Questions

Question No. 1

See the graph of f(x)=−x²+7x−6.

Answer :

a) Estimate the shaded area in the diagram by counting the shaded squares.

b) Use a definite integral to calculate the shaded area.

Question No. 2

Below is the graph of f(x)=½x²−2.

Answer :

a) Estimate the shaded area in the diagram by counting the shaded squares.

b) Use Simpson’s rule to calculate the shaded area.

Question No. 3

Evaluate each integral.

Answer :

a) ∫x³ dx

b) ∫x⁴ dx

c) ∫x dx

d) ∫x⁷ dx

e) ∫√x dx

f) ∫³√(x²) dx

Question No. 4

Now check your answers to question 1 by finding the derivative of each answer. Do your answers show that integration and differentiation are the reverse of each other? If not, check your calculations carefully.

Answer :

Question No. 5

Evaluate the following integrals.Use differentiation to check your answers

Question No. 6

Evaluate the following. Use differentiation to check your answers.

Answer :

b) ∫2x⁷ dx

c) ∫(x − 4)² dx

a) ∫⅓ dx

d) ∫(sin x + 2 cos x + 3) dx

e) ∫(2 sin x + x³⁄₄) dx

f) ∫(1⁄x³ − cos x) dx

g) ∫(5√x − ⁴√x⁵ − 15) dx

h) ∫[x²(1 ⁄ x² − 3 ⁄ x⁶ + 9)] dx

i) ∫(x⁶ + ¹²⁄₅ x⁵ + ²⁄₃ x⁴) dx

Question No. 7

Use integration by substitution to evaluate the following.

Answer :

a) ∫(2 sin 2x) dx (Hint: Let u = 2x.)

b) ∫x²(x³ + 1) dx (Hint: Let u = x³ + 1.)

c) ∫sin x × cos x dx (Hint: Let u = sin x.)

d) ∫cos 5x dx (Hint: Let u = 5x.)

e) ∫(x³ + x)⁹(3x² + 1) dx (Hint: Let u = x³ + x.)

f) ∫sin⁵ x × cos x dx (Hint: Let u = sin x.)```

Question No. 8

Use integration by partial fractions to evaluate the following.

Answer :

a) ∫4/(x² + 4x + 3) dx

b) ∫x/(x² − 6x + 8) dx

c) ∫(x + 14)/(x² + 7x + 10) dx

d) ∫(3x − 7)/(x² − 4x + 3) dx

e) ∫(5x − 8)/(x² − 2x − 8) dx

f) ∫(−4x + 12)/(x² + 6x − 7) dx

Question No. 9

The graph of f(x)=−2x+6 is shown on the right.

Answer :

a) Use the formula for the area of a triangle to calculate the shaded area in the diagram.

b) Use a definite integral to calculate the shaded area.

Question No. 10

The graph of f(x)=−x³+4x²−3x is shown below.Use definite integrals to calculate the sum of the shaded areas.

Answer :

Question No. 11

Calculate the areas enclosed between the graph of each function, the x-axis and the given values of x (shaded area on one side only).

Answer :

a) f(x)=2x+1 (x=1 and x=3)

b) f(x)=x²−x−12 (x=−3 and x=4)

c) f(x)=x³+8 (x=−2 and x=0)

d) f(x)=6 (x=2 and x=6)

Question No. 12

In the diagram, v is the velocity (in metres per second) of an object after t seconds. It is described by the function v(t) = 0.5t.

Answer :

a) Give the initial velocity of the object.

b) What is the velocity of the object when t equals 6 seconds?

c) What does the shaded area in the diagram represent?

d) Use a definite integral to calculate the shaded area in the diagram and include the correct unit of measurement with your answer.

Question No. 13

A football on the ground is kicked into the air. The trajectory (path) of the football is described by f(x), where x is time (seconds) after the ball was kicked, and f(x) is the height (metres) of the football above the ground. It is further known that the gradient of the tangent to the curve of f(x) is 12(1 − x).

Answer :

a) Use the above information to determine f(x). (Hint: Remember that the gradient of the tangent to any curve is equal to the derivative of the function that defines the curve.)

b) Calculate the maximum height of the football above the ground.

c) How long will it take for the football to reach the ground again?

Question No. 14

An object moves in a straight line. Its acceleration is 6t², where the time t is measured in seconds and its distance from its starting point is measured in metres. When t = 0, the object is stationary (it is not moving).

Answer :

a) Express the velocity of the object in terms of t.

b) Express the distance of the object from its starting point in terms of t.

c) How far is the object from its starting point after three seconds?

Question No. 15

The first diagram shows the area bounded by the straight line f(x) = x, the x-axis and the line x = 2. If this shaded area is rotated through a complete revolution of 360° about the x-axis, the resulting shape is a cone, as shown in the second diagram. The volume of this cone is given by the formula V = ∫₀² π [f(x)]² dx.

Answer :

a) Use this formula to calculate the volume of the cone.

b) Use the following formula to calculate the volume of the cone and check your answer to question a): V = ¹⁄₃ π r² h.

Question No. 16

Exercise 12.7

Answer :

a) Need to do it manually

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