Quiz about Integration Worksheet A

Question 1

Evaluate $$\int_{1}^{3} (4x - 1) \, dx$$ .

Accepted Answer

5

Answer

10

Answer

15

Answer

20

Question 2

Evaluate $$\int_{0}^{1} (3x^2 + 2) \; dx$$ .

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 3

Evaluate $$\int_{2}^{3} (x - x^3) \, dx$$ .

Accepted Answer

-1/4

Answer

-5/4

Answer

1/4

Answer

5/4

Question 4

Evaluate $$\int_{1}^{2} (3x + 1)^2 \, dx$$ .

Accepted Answer

$$\frac{37}{3}$$

Answer

$$\frac{19}{3}$$

Answer

$$\frac{28}{3}$$

Answer

$$\frac{46}{3}$$

Question 5

Evaluate $$\int_{1}^{2} (x^2 - 8x - 3) \, dx$$ .

Accepted Answer

$$-\frac{26}{3}$$

Answer

$$-\frac{25}{3}$$

Answer

$$-\frac{27}{3}$$

Answer

$$-\frac{28}{3}$$

Question 6

Evaluate $$\int_{-2}^{4} (8 - 4x + 3x^2) \, dx$$ .

Accepted Answer

56

Answer

48

Answer

64

Answer

72

Question 7

Evaluate $$\int_{1}^{4} (x^3 - 2x - 7) \, dx$$ .

Accepted Answer

-65

Answer

-60

Answer

-70

Answer

-75

Question 8

Evaluate $$\int_{-2}^{-1} (5 + x^2 - 4x^2) \; dx$$ .

Accepted Answer

$$-\frac{7}{3}$$

Answer

$$\frac{7}{3}$$

Answer

$$-\frac{8}{3}$$

Answer

$$\frac{8}{3}$$

Question 9

Evaluate $$\int_{-1}^{2} (x^4 + 6x^2 - x) \, dx$$ .

Accepted Answer

5.25

Answer

6.25

Answer

7.25

Answer

8.25

Question 10

Given that $$\int_{1}^{4} (3x^2 + ax - 5) \, dx = 18$$ , find the value of the constant $$a$$ .

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 11

Given that $$\int_{-1}^{k} (3x^2 - 12x + 9) \, dx = 16$$ , find the value of the non-zero constant $$k$$ .

Accepted Answer

k = 3

Answer

k = 2

Answer

k = 4

Answer

k = 5

Question 12

Evaluate $$\int_{1}^{3} \left(2 - \frac{1}{x^2}\right) \, dx$$ .

Accepted Answer

3

Answer

2

Answer

4

Answer

5

Question 13

Evaluate $$\int_{-2}^{-1} \left(6x + \frac{4}{x}\right) \, dx$$ .

Accepted Answer

-10 - 2ln(2)

Answer

10 + 2ln(2)

Answer

-10 + 2ln(2)

Answer

10 - 2ln(2)

Question 14

Evaluate $$\int_{1}^{4} (3x^{\frac{1}{2}} - 4) \, dx$$ .

Accepted Answer

6

Answer

5

Answer

7

Answer

8

Question 15

Evaluate $$\int_{-1}^{2} \frac{4x^2 - x}{2x} \, dx$$ .

Accepted Answer

4.5

Answer

3.5

Answer

5.5

Answer

6.5

Question 16

Evaluate $$\int_{1}^{8} (x - x^{\frac{1}{3}}) \, dx$$ .

Accepted Answer

24

Answer

21

Answer

25

Answer

27

Question 17

Evaluate $$\int_{2}^{3} \frac{1 - 6x^2}{3x^2} \, dx$$ .

Accepted Answer

-1/3

Answer

1/3

Answer

-1

Answer

1

Question 18

The diagram shows the curve with the equation $$y = 3x^2 - 6x + 7$$ . What is the area of the shaded region enclosed by the curve, the x-axis and the lines $$x = 1$$ and $$x = 3$$ ?

Accepted Answer

8 square units

Answer

4 square units

Answer

12 square units

Answer

16 square units

Question 19

The diagram shows the curve with the equation $$y = 4 - x^2$$ . What are the coordinates of the points where the curve crosses the x-axis?

Accepted Answer

(2, 0) and (-2, 0)

Answer

(0, 2) and (0, -2)

Answer

(4, 0) and (-4, 0)

Answer

(0, 4) and (0, -4)

Question 20

In each part of this question, sketch the given curve and find the area of the region enclosed by the curve and the x-axis. a) $$6x - 3x^2$$

Accepted Answer

The area is 18 square units.

Answer

The area is 9 square units.

Answer

The area is 12 square units.

Answer

The area is 27 square units.

Question 21

In each part of this question, sketch the given curve and find the area of the region enclosed by the curve and the x-axis. b) $$-x^2 + 4x - 3$$

Accepted Answer

6 square units

Answer

8 square units

Answer

4 square units

Answer

10 square units

Question 22

In each part of this question, sketch the given curve and find the area of the region enclosed by the curve and the x-axis. c) $$4 - 3x - x^2$$

Accepted Answer

The area is 14 square units.

Answer

The area is 10 square units.

Answer

The area is 12 square units.

Answer

The area is 16 square units.

Question 23

In each part of this question, sketch the given curve and find the area of the region enclosed by the curve and the x-axis. What is the area of the region enclosed by the curve $$y = 2x^{1/2} - x$$ and the x-axis?

Accepted Answer

1.33 square units

Answer

0.67 square units

Answer

2 square units

Answer

2.67 square units

Question 24

Sketch the curve with the equation $$y = x^2 + 4x$$ . Find the area of the region enclosed by the curve, the x-axis and the line $$x = 2$$ .

Accepted Answer

8 square units

Answer

10 square units

Answer

12 square units

Answer

14 square units

Question 25

The diagram shows the curve with the equation y = $$x^2 + 2x - 15$$ . What are the coordinates of the points where the curve crosses the x-axis?

Accepted Answer

(-5, 0) and (3, 0)

Answer

(-3, 0) and (5, 0)

Answer

(5, 0) and (-3, 0)

Answer

(3, 0) and (-5, 0)

Question 26

The diagram shows the curve with the equation $$y = x^2 + 2x - 15$$ . c) State the area of the shaded region enclosed by the curve, the y-axis and the positive x-axis.

Accepted Answer

0 square units

Answer

15 square units

Answer

30 square units

Answer

45 square units

Question 27

The diagram shows the curve with the equation $$y = x^3 - 5x^2 + 6x$$ . What are the coordinates of the points where the curve crosses the x-axis?

Accepted Answer

(0, 0), (2, 0), (3, 0)

Answer

(1, 0), (2, 0), (3, 0)

Answer

(0, 0), (1, 0), (3, 0)

Answer

(0, 0), (2, 0), (4, 0)

Question 28

The diagram shows the curve with the equation y = $$x^3 - 5x^2 + 6x$$ . b) Show that the total area of the shaded regions enclosed by the curve and the x-axis is $$3 \frac{1}{12}$$ .

Accepted Answer

The total area is 3 1/12.

Answer

The total area is 4.

Answer

The total area is 2 1/2.

Answer

The total area is 5.

Question 29

The diagram shows the curve y = $$x^2 - 3x + 4$$ and the straight line y = $$x + 1$$ . b) Find the area of the shaded region enclosed by the curve and the line.

Accepted Answer

3 square units

Answer

2 square units

Answer

4 square units

Answer

5 square units

Question 30

In each part of this question sketch the given curve and line on the same set of coordinate axes and find the area of the region enclosed by the curve and line. a) $$y = 9 - x^2$$ and $$y = 6 - 2x$$

Accepted Answer

C) 13.5 square units

Answer

A) 4.5 square units

Answer

B) 9 square units

Answer

D) 18 square units

Question 31

In each part of this question sketch the given curve and line on the same set of coordinate axes and find the area of the region enclosed by the curve and line. b) $$y = x^2 - 4x + 4$$ and $$y = 16$$

Accepted Answer

Area = 12 square units

Answer

Area = 8 square units

Answer

Area = 16 square units

Answer

Area = 10 square units

Question 32

In each part of this question sketch the given curve and line on the same set of coordinate axes and find the area of the region enclosed by the curve and line. c) $$y = x^2 - 5x - 6$$ and $$y = x - 11$$

Accepted Answer

Area = 22 square units

Answer

Area = 18 square units

Answer

Area = 20 square units

Answer

Area = 24 square units

Question 33

1a. Express f(x) in the form $$a(x + b)^2 + c$$ , stating the values of the constants a, b, and c for the function $$f(x) \equiv 3 + 4x - x^2$$ .

Accepted Answer

a = -1, b = -2, c = 7

Answer

a = -1, b = 2, c = 7

Answer

a = 1, b = 2, c = -7

Answer

a = 1, b = -2, c = -7

Question 34

Evaluate $$\int_{1}^{2} \frac{8}{x^3} \, dx$$ .

Accepted Answer

-4

Answer

-2

Answer

2

Answer

4

Question 35

3a. Find an equation of the normal to the curve at P for the curve $$y = 5x - 2x^2$$ .

Accepted Answer

y = -2x + 3

Answer

y = 2x - 3

Answer

y = -2x - 3

Answer

y = 2x + 3

Question 36

3b. Show that the area of the shaded region is $$\frac{5}{3}$$ for the curve $$y = 5x - 2x^2$$ .

Accepted Answer

The area of the shaded region is $$\frac{5}{3}$$ .

Answer

The area of the shaded region is $$\frac{4}{3}$$ .

Answer

The area of the shaded region is $$\frac{7}{3}$$ .

Answer

The area of the shaded region is $$\frac{8}{3}$$ .

Question 37

4a. Find the coordinates of the point P where C crosses the x-axis for the curve $$y = \frac{4 - x^2}{x}$$ .

Accepted Answer

(2, 0)

Answer

(-2, 0)

Answer

(0, 4)

Answer

(4, 0)

Question 38

4b. Find the coordinates of the point Q for the curve $$y = \frac{4 - x^2}{x}$$ .

Accepted Answer

(2, 0)

Answer

(0, 4)

Answer

(1, 3)

Answer

(-2, -4)

Question 39

4c. Show that the area of the shaded region enclosed by C and l is $$\frac{1}{2}$$ for the curve $$y = \frac{4 - x^2}{x}$$ .

Accepted Answer

True

Answer

False

Question 40

The shaded region is bounded by the curve, the x-axis and the lines x = 1 and x = 4. Use the trapezium rule with all the values in your table to show that the area of the shaded region is approximately 4 3/8.

Accepted Answer

4 3/8

Answer

4 1/4

Answer

4 1/2

Answer

4 5/8

Question 41

With the aid of a sketch diagram, determine if the true area is more or less than $$4\frac{3}{8}$$ .

Accepted Answer

More than 4 $$\frac{3}{8}$$

Answer

Less than 4 $$\frac{3}{8}$$

Question 42

Sketch the curve $$y = x(3x + 2)$$ showing the coordinates of any points of intersection with the coordinate axes.

Accepted Answer

The curve intersects the x-axis at (0,0) and (-2/3,0) and the y-axis at (0,0).

Answer

The curve intersects the x-axis at (0,0) and (2/3,0) and the y-axis at (0,0).

Answer

The curve intersects the x-axis at (0,0) and (-2,0) and the y-axis at (0,0).

Answer

The curve intersects the x-axis at (0,0) and (3,0) and the y-axis at (0,0).

Question 43

Use the trapezium rule with 4 intervals of equal width to estimate the area bounded by the curve, the x-axis and the line $$x = 2$$ .

Accepted Answer

6.8

Answer

5.2

Answer

7.5

Answer

8.1

Question 44

Find this area exactly using integration.

Accepted Answer

B. 1

Answer

A. 0

Answer

C. 2

Answer

D. 3

Question 45

Hence, find the percentage error in the estimate made in part b.

Accepted Answer

1%

Answer

0.5%

Answer

1.5%

Answer

2%

Question 46

Use the trapezium rule with the stated number of intervals of equal width to estimate the area of the region enclosed by the given curve, the x-axis and the given ordinates. $$y = \frac{3}{2x+1}$$ , $$x = 4$$ , $$x = 6$$ , 2 intervals.

Accepted Answer

3.0

Answer

2.5

Answer

3.5

Answer

4.0

Question 47

Use the trapezium rule with the stated number of intervals of equal width to estimate the area of the region enclosed by the given curve, the x-axis and the given ordinates. $$y = \frac{1}{x} (x^2 + 9)$$ , $$x = 0$$ , $$x = 3$$ , 3 intervals.

Accepted Answer

11.5

Answer

9.5

Answer

10.5

Answer

12.5

Question 48

Use the trapezium rule with the stated number of intervals of equal width to estimate the area of the region enclosed by the given curve, the x-axis and the given ordinates. $$y = x^2 \sin x$$ , $$x = 0$$ , $$x = \pi$$ , 4 intervals. What is the estimated area?

Accepted Answer

5.8696

Answer

6.2832

Answer

7.1234

Answer

8.4567

Question 49

Use the trapezium rule with the stated number of intervals of equal width to estimate the area of the region enclosed by the given curve, the x-axis and the given ordinates. $$y = \sqrt{2x+5}$$ , $$x = -2$$ , $$x = 2$$ , 4 intervals.

Accepted Answer

9.0

Answer

8.5

Answer

9.5

Answer

10.0

Question 50

Use the trapezium rule with the stated number of equally-spaced ordinates to estimate the area of the region enclosed by the given curve, the x-axis and the given ordinates. $$y = 3^x$$ , $$x = 0$$ , $$x = 3$$ , 4 ordinates.

Accepted Answer

12.0

Answer

10.5

Answer

14.5

Answer

16.0

Question 51

Use the trapezium rule with the stated number of equally-spaced ordinates to estimate the area of the region enclosed by the given curve, the x-axis and the given ordinates. $$y = \sin (\lg x)$$ , $$x = 2$$ , $$x = 2.4$$ , 3 ordinates.

Accepted Answer

0.20

Answer

0.15

Answer

0.25

Answer

0.30

Question 52

Use the trapezium rule with the stated number of equally-spaced ordinates to estimate the area of the region enclosed by the given curve, the x-axis and the given ordinates. $$y = \frac{x}{x^2 + 2}$$ , $$x = 0$$ , $$x = 0.5$$ , 6 ordinates.

Accepted Answer

0.052

Answer

0.048

Answer

0.056

Answer

0.060

Question 53

Use the trapezium rule with the stated number of equally-spaced ordinates to estimate the area of the region enclosed by the given curve, the x-axis and the given ordinates. $$y = \sqrt{\cos \left(\frac{1}{2} x\right)}$$ , $$x = 0$$ , $$x = \frac{2\pi}{3}$$ , 5 ordinates. What is the estimated area?

Accepted Answer

0.7

Answer

0.5

Answer

0.6

Answer

0.8

Question 54

The diagram shows the finite region, R, which is bounded by the curve y = 2 - 3x^{- $$\frac{1}{2}$$ }, the x-axis and the lines x = 3 and x = 7. Use the trapezium rule with 5 intervals of equal width to estimate the area of R.

Accepted Answer

14.7

Answer

10.5

Answer

12.3

Answer

16.2

Question 55

The diagram shows the finite region, R, which is bounded by the curve y = 2 - 3x^{-}, the x-axis and the lines x = 3 and x = 7. Use integration to find the exact area of R.

Accepted Answer

6

Answer

4

Answer

5

Answer

7

Question 56

The diagram shows the curve y = $$\sin x^2$$ , 0 \leq x \leq 1 and the lines x = 1 and y = $$\sin 1$$ . Use the trapezium rule with 5 strips of equal width to estimate the area bounded by the curve y = $$\sin x^2$$ , the x-axis and the line x = 1, giving your answer to 4 decimal places.

Accepted Answer

0.3300

Answer

0.3100

Answer

0.3200

Answer

0.3400

Question 57

The finite region R is bounded by the curve y = $$(1 + \frac{x}{2})^6$$ , the coordinate axes and the line x = 0.5. Use your expression in a and integration to find an estimate for the area of R.

Accepted Answer

0.2

Answer

0.1

Answer

0.3

Answer

0.4

Question 58

The finite region R is bounded by the curve y = $$(1 + \frac{x}{2})^6$$ , the coordinate axes and the line x = 0.5. Use the trapezium rule with 6 equally-spaced ordinates to find another estimate for the area of R.

Accepted Answer

0.7

Answer

0.5

Answer

0.6

Answer

0.8

Question 59

The diagram shows the curve y = $$x^2 + \frac{16}{x}$$ for x > 0. What are the coordinates of the stationary point on the curve?

Accepted Answer

(2, 12)

Answer

(3, 15)

Answer

(4, 16)

Answer

(1, 10)

Question 60

Evaluate $$\int_{1}^{4} \frac{2}{x^2} \, dx$$ .

Accepted Answer

1

Answer

2

Answer

3

Answer

4

Question 61

Evaluate $$\int_{0}^{2} (x - 3)^2 \, dx$$ .

Accepted Answer

8

Answer

4

Answer

12

Answer

16

Question 62

Use the trapezium rule with three intervals of equal width to estimate the area of the shaded region bounded by the curve y = $$\sqrt{x^2 + 4}$$ , the x-axis, and the lines x = 0 and x = 6.

Accepted Answer

19.5

Answer

18.0

Answer

20.0

Answer

21.5

Question 63

State, with a reason, whether your answer to part a is an under-estimate or an over-estimate of the true area.

Accepted Answer

The answer to part a is an under-estimate of the true area because it does not account for all variables.

Answer

The answer to part a is an over-estimate of the true area because it includes additional variables.

Answer

The answer to part a is an accurate estimate of the true area.

Question 64

Find the value of $$f(2)$$ for $$f(x) = 3x^{\frac{1}{2}} - x^{\frac{1}{3}}$$ , giving your answer in the form $$k\sqrt{2}$$ where $$k$$ is an exact fraction.

Accepted Answer

$$\frac{5}{2}\sqrt{2}$$

Answer

$$\frac{7}{3}\sqrt{2}$$

Answer

$$\frac{4}{3}\sqrt{2}$$

Answer

$$\frac{3}{2}\sqrt{2}$$

Integration Worksheet A

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