SCIENCE AND MATH CONCEPTS

Exercise 8.1  Question 1. Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4, and the x-axis. Solution: The curve y² = x is a parabola with vertex at origin.Axis of x is the line of symmetry, which is the axis of parabola. The area of the region bounded by the curve, x = 1, x=4 and the x-axis. Area LMQP Question 2. Find the area of the region bounded by y² = 9x, x = 2, x = 4 and x-axis in the first quadrant Solution: The given curve is y² = 9x, which is a parabola with vertex at (0, 0) and axis along x-axis. It is symmetrical about x-axis, as it contains only even powers of y. x = 2 and x = 4 are straight lines parallel toy-axis at a positive distance of 2 and 4 units from it respectively. ∴ Required area = Area ABCD Question 3. Find the area of the region bounded by x² = 4y, y = 2, y = 4 and the y-axis in the first quadrant. Solution: The given curve x² = 4y is a parabola with vertex at (0,0). Also since it contains only even powers of x,it is symmetric

Exercise 7.8 Evaluate the following definite integral as limit of sums.   Question 1. Solution: on comparing we have Question 2 Solution: on comparing we have f(x) = x+1, a = 0, b = 5 and nh = b-a = 5-0 = 5 Question 3. Solution: compare we have Question 4. Solution: compare we have f(x) = x²-x and a = 1, b = 4 Question 5. Solution: compare we have Question 6. Solution: let f(x) = x + e 2x , a = 0, b = 4 and nh = b – a = 4 – 0 = 4 Exercise 7.9 Evaluate the definite integrals in Exercise 1 to 20. Question 1. Solution: Question 2. Solution: Question 3. Solution: Question 4. Solution: Question 5. Solution: Question 6. Solution: Question 7. Solution: Question 8. Solution: Question 9. Solution: Question 10. Solution: Question 11. Solution: Question 12. Solution: Question 13. Solution: Question 14. Solution: Question 15. Solution: let x² = t ⇒ 2xdx = dt when x = 0, t = 0 & when x = 1,t = 1 Question 16. Solution: Question 17. Solution: Question 18. Solution: Question 19. Solution: Question