Calculus  Integration  Areas.
Test Yourself 2.
The questions on this page address: 
1. Areas from the xaxis. 
2. Areas from the yaxis. 
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 xaxis.  1. Find the exact area bounded by the curve y = x^{2}  2, the x axis and the lines x = 3 and x = 5. Answer.Area is 28.67 u^{2}. 
2.  
3. Find the area of the region bounded by the curve y = 3x^{2}(5  x) and the x axis.
Answer.Area is 156.25 u^{2}. 

4.  
5. Sketch on the number plane and label a function whose area between the curve and the xaxis 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 u^{2}. What is the value of k? Answer.k = 1/3. 

From the yaxis.  7. (i) Sketch the curve y = 4x  x^{2}.
(ii) Determine the area between the parabola and the y axis between y = 0 and y = 4. Answer.Area is 8/3 = 2.67 u^{2}. 
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  x^{2}.
Answer.Area = 36 u^{2}. 
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 

12. The diagram below shows the two parabolas y = x^{2} + x + 1 and y = 2x^{2}  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 u^{2}. 

13. Calculate the area of the region enclosed between the curves f(x) = x + 1 and g(x) = x^{2}  x  2. Answer.Area = 32/3 u^{2}. 

14. Calculate the area of the region bounded by
f(x) = x^{2}, g(x) = x^{2} (for x > 0), the x axis and the line x = 3.
Answer.Area = 1 u^{2}. 

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.

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 . 