Calculus - Integration - Areas.
Test Yourself 2.
The questions on this page address: |
1. Areas from the x-axis. |
2. Areas from the y-axis. |
3. Areas between 2 curves - two points of intersection. |
4. Areas between 2 curves - one point of intersection. |
5. Integrals only!!! |
6. Interpreting diagrams. |
From the x-axis. | 1. Find the exact area bounded by the curve y = x2 - 2, the x axis and the lines x = 3 and x = 5. Answer.Area is 28.67 u2. |
2. | |
3. Find the area of the region bounded by the curve y = 3x2(5 - x) and the x axis.
Answer.Area is 156.25 u2. |
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4. | |
5. Sketch on the number plane and label a function whose area between the curve and the x-axis can be represented by the statement: | |
6. The graph of a function y = k (x + 1)3 is shown below for the domain [-3, 1]. The value of k is a positive constant.
The area of the shaded region is 8/3 u2. What is the value of k? Answer.k = 1/3. |
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From the y-axis. | 7. (i) Sketch the curve y = 4x - x2.
(ii) Determine the area between the parabola and the y axis between y = 0 and y = 4. Answer.Area is 8/3 = 2.67 u2. |
8. | |
9. | |
Between 2 curves - 2 points of intersection. | 10. Find the area of the region defined by the inequalities y ≥ -5 and by y ≤ 4x - x2.
Answer.Area = 36 u2. |
11. The curves y = (x - 1)2 and x + y = 3 intersect at A and B as shown in the diagram.
(i) Verify using algebra that one of the points of intersection has coordinates (2, 1). (ii) Hence find the area enclosed by the curve |
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12. The diagram below shows the two parabolas y = x2 + x + 1 and y = 2x2 - x - 2.
(i) Show the two parabolas intersect at x = -1 and x = 3. (ii) Find the area enclosed between the two parabolas. Answer.(ii) Area = 32/3 u2. |
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13. Calculate the area of the region enclosed between the curves f(x) = x + 1 and g(x) = x2 - x - 2. Answer.Area = 32/3 u2. |
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14. Calculate the area of the region bounded by
f(x) = x2, g(x) = x-2 (for x > 0), the x axis and the line x = 3.
Answer.Area = 1 u2. |
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Between 2 curves - one point of intersection. | 15. |
16. | |
Integrals! | 17. The diagram above shows the graph of a function y = f(x). The function consists of two quadrants of a circle (AB and DE) a straight line segment BC and a horizontal line CD.
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18. | |
Interpreting diagrams. | 19. The function y = f(x) is drawn in the diagram below.
Evaluate |
20.
The diagram above shows the graph of y = f(x) Evaluate . |