Calculus - Integration

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.

6. Interpreting diagrams.

From the x-axis.	1. Find the area enclosed between the parabola y = x² - 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 = -x² + 8x -12 and the x axis. Hint.Draw a quick sketch of the parabola and add the slice. Answer.Area = 32/3 u².
	3. (i) Sketch the curve for 0 ≤ x ≤ 2. (ii) Find the area between the above curve and the x - axis between x = 0 and x = 1. Answer.Area = 1/3 u^.2
	4. (i) Sketch the curve y = x² - x + 6 for 0 ≤ x ≤ 4. (ii) Find the area bounded by the curve, the coordinate axes and the line x = 4. Answer.Area = 112/3 u^.2
	5. (i) Sketch the function f(x) = \|x - 4\|. (ii) Hence or otherwise, evaluate . Answer.Area = 10 u².
	6. Find the exact area enclosed between the parabola y = 2 - (x - 2)² and the x-axis. Answer.Area = 32/3 u^.2
	7. (i) Sketch the curve y = x² - 3x for [-1, 4] (ii) Hence or otherwise, find the area between y = x² - 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 u²
From the y-axis.	9. (i) Sketch the curve y = x² + 2x. (ii) Determine the area between the parabola and the y axis between y = 0 and y = 3. Answer.Area = 1.67 u²
	10. Find the area bounded by y = x³, the y axis and y = 0 and y = 8. Answer.Area = 12 u²
Between 2 curves - 2 points of intersection.	11. (i) Sketch the two curves y = 2x and y = 6x - x². (ii) Determine the shaded area between the two curves. Hint.Draw a quick sketch of the line and the parabola and add a vertical slice between them. Answer.Area = 10.67 u^.2
	12. (i) Sketch the two curves f(x) = x² and g(x) = 3x - 2. (ii) Find the two points of intersection between these two curves. (iii) Find the area bounded by f(x) and g(x). Answer.Area = 1/6 u^.2
	13. Calculate the area of the region bounded by f(x) = x², 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 = x² + 1 at the point where x = 2. (ii) Sketch the parabola and its tangent as well as the line x = 2. (iii) Find the area enclosed by the parabola, the tangent and the coordinate axes. Problem.We cannot calculate the distance between the curves as "top - bottom curve" as the bottom curve changes from the tangent to the x-axis 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 = 2x² and y = 12 - x² and the x-axis from x = 0 to x = 2√3. (i) Find the coordinates of point X where the two parabolas meet in the first quadrant. (ii) Calculate the area of the region OXY correct to two decimal places. Answer.(i) (2, 4). (ii) Area = 13.19 u².
	16. The curves y = x² and y = 4x - x² intersect at the origin and at the point A. (i) Show that point A has coordinates (2, 4). (ii) Hence find the area enclosed between the two curves. Answer.Area = 8/3 u^.2
Integrals!	17. Evaluate . Answer. I = 0.
	18. (i) Sketch the graph of y = x² - x - 2 for -2 ≤ x ≤ 4. (ii) Evaluate . (iii) Calculate the area between the curve and the x axis from x = 1 to x = 4. (iv) Compare your answers for parts (ii) and (iii) and explain the difference. Answer.(ii) -7.5. (iii) Area = 59/6.
Interpreting diagrams.	19. Part of the graph of the function y = -x² + ax + 12 is shown in the diagram below. If the shaded area is 45 u², 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.