Application of Integrals

Application of Integrals uses the definite integral, defined as the limit of a sum, to find the areas of regions bounded by curves, a problem that cannot be solved with the elementary formulas of geometry. In this chapter you will learn that the area bounded by a curve y = f(x), the x-axis and the ordinates x = p and x = q is the integral of f(x) dx from p to q, obtained by adding the areas of thin vertical strips. Similarly, the area bounded by x = g(y), the y-axis and the lines y = p and y = q is found using horizontal strips. You will apply these ideas to simple curves in standard form, namely straight lines, circles, parabolas and ellipses, and see why the area of an ellipse with semi-axes a and b is pi ab. You will learn how to handle regions that lie below the x-axis, where the integral is negative and its absolute value must be taken, and how symmetry of a curve about an axis halves the work.

Detailed Worksheet: Application of Integrals Section A - Definitions (10 marks) 1. Write the formula for the area bounded by the curve y = f(x), the x-axis and the lines x = p and x = q, where f(x) >= 0. What is the elementary strip used? (2 marks) 2. Write the formula for the area bounded by the curve x = g(y), the y-axis and the lines y = p and y = q. (2 marks) 3. If a region lies below the x-axis, why is its definite integral negative? How is the area then calculated? (2 marks) 4. Find the area bounded by y = x, the x-axis and the line x = 2. (2 marks) 5. Using symmetry, explain why the area enclosed by the circle x^2 + y^2 = a^2 is four times the area in the first quadrant. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the area of the region bounded by y^2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant. (3 marks) 7. Using integration, find the area enclosed by the circle x^2 + y^2 = 16. (3 marks) 8. Find the area of the region bounded by the ellipse x^2/16 + y^2/9 = 1. (3 marks) 9. Find the area of the region bounded by x^2 = 4y, y = 2, y = 4 and the y-axis in the first quadrant. (3 marks) 10. Find the area bounded by the line y = 3x + 2, the x-axis and the ordinates x = -1 and x = 1. (3 marks) Section C - Diagrams (10 marks) 11. Sketch the region bounded by the ellipse x^2/a^2 + y^2/b^2 = 1, shade the part in the first quadrant, show a vertical strip and write the integral for the total area. (4 marks) 12. Sketch the parabola y^2 = 4ax and its latus rectum x = a. Shade the region enclosed by them and show a vertical strip. (3 marks) 13. Sketch the curve y = x^2 and the line y = 4. Shade the region enclosed between them and show a horizontal strip. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Find the area of the region bounded by the curve y = x^2 and the line y = 4, using horizontal strips and the symmetry of the region. (5 marks) 15. Find the area of the region bounded by the parabola y^2 = 4ax and its latus rectum. Hence find the area when a = 3. (5 marks) 16. Using integration, prove that the area of the ellipse x^2/a^2 + y^2/b^2 = 1 is pi ab. Deduce the area of a circle of radius r from this result. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw a neat sketch of every region before integrating and show the limits clearly.