Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

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.
  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.
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
y = (x - 1)2 and the lines x + y = 3 and x = 3.

  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.
  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.
  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.
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.

(i) Find the area between the function and the x axis for 0 ≤ x ≤ 8.

(ii) Hence evaluate .

(iii) For what values of x in the domain
0 ≤ x ≤ 8 is the function not differentiable?

 
  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)
over the domain 0≤ x≤ 12.

Evaluate .