Calculus  Integration  Areas.
Test Yourself 1.
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 area enclosed between the parabola y = x^{2}  6x + 8 and the two coordinate axes.
Hint.Draw a quick sketch of the parabola and add the slice. Answer.Area = 20/3 u^{.2} 
2. Find the area enclosed between the parabola y = x^{2} + 8x 12 and the x axis. Hint.Draw a quick sketch of the parabola and add the slice. Answer.Area = 32/3 u^{2}. 

3. (i) Sketch the curve for 0 ≤ x ≤ 2.
Answer.Area = 1/3 u^{.2} 

4. (i) Sketch the curve y = x^{2}  x + 6 for 0 ≤ x ≤ 4.
Answer.Area = 112/3 u^{.2} 

5. (i) Sketch the function f(x) = x  4.
Answer.Area = 10 u^{2}. 

6. Find the exact area enclosed between the parabola y = 2  (x  2)^{2} and the xaxis. Answer.Area = 32/3 u^{.2} 

7. (i) Sketch the curve y = x^{2}  3x for [1, 4]
(ii) Hence or otherwise, find the area between y = x^{2}  3x and the x axis between x = 1 and x = 4. Answer.Area = 8.16 u^{.2} 

8. (i) Sketch the function y = 2x  6  4 in the domain [1, 6].
(ii) Hence evaluate the area between the curve and the x axis between x = 1 and x = 6. Answer.Area = 13 u^{2} 

From the yaxis.  9. (i) Sketch the curve y = x^{2} + 2x.
(ii) Determine the area between the parabola and the y axis between y = 0 and y = 3. Answer.Area = 1.67 u^{2} 
10. Find the area bounded by y = x^{3}, the y axis and y = 0 and y = 8. Answer.Area = 12 u^{2} 

Between 2 curves  2 points of intersection. 
11. (i) Sketch the two curves y = 2x and y = 6x  x^{2}.
Answer.Area = 10.67 u^{.2} 
12. (i) Sketch the two curves f(x) = x^{2} and g(x) = 3x  2.
Answer.Area = 1/6 u^{.2} 

13. Calculate the area of the region bounded by
f(x) = x^{2}, g(x) = x^{2} (for x > 0) and the line x = 3. Answer.Area = 26/3 u^{.2} 

Between 2 curves  one point of intersection.  14. (i) Find the equation of the tangent to the parabola y = x^{2} + 1 at the point where x = 2. Problem.We cannot calculate the distance between the curves as "top  bottom curve" as the bottom curve changes from the tangent to the xaxis at x = 0.75. Answer.(i) Tangent is y = 4x  3. (iii) Area = 27/34 u^{.2} 
15. The shaded region OXY in the diagram is bounded by the parabolas y = 2x^{2} and y = 12  x^{2} and the xaxis from x = 0 to x = 2√3.
Answer.(i) (2, 4). (ii) Area = 13.19 u^{2}. 

16. The curves y = x^{2} and y = 4x  x^{2} intersect at the origin and at the point A.
Answer.Area = 8/3 u^{.2} 

Integrals!  17. Evaluate . Answer. I = 0. 
18. (i) Sketch the graph of y = x^{2}  x  2 for 2 ≤ x ≤ 4.
Answer.(ii) 7.5. (iii) Area = 59/6. 

Interpreting diagrams.  19. Part of the graph of the function y = x^{2} + ax + 12 is shown in the diagram below.
If the shaded area is 45 u^{2}, find the value of a. Answer.a = 4. 
20. The diagram below illustrates the function y = f(x).
(i) Evaluate . (ii) Find two values of a such that . Answer.(i) Integral = 5.5.(ii) Values for a are 2 and 3. 