APPLICATION OF INTEGRATION

1. APPLICATION OF INTEGRATION

2. The integration can be used to determine the area bounded by the plane curves, arc lengths volume and surface area of a region bounded by revolving a curve about a line.

3. I. AREA OF THE PLANEREGION • We know that the area bounded by a Cartesian curve y = f(x), x – axis, between lines x = a & x = b given by

4. Y (0, 0) x

5. r=f()  = c2 Y  = c1 X 0 r  o Area=(1/2)r2 

6. Example 1: • To find the area lying between the parabola y =4x – x2 and the line y = x. • The required area = ( The area bounded by the parabola y =4x – x2, the x – axis, in between lines x = 0 and x = 3) – the area bounded by y = x, x – axis, in between lines x = 0 and x = 3.

7. Y y = x (3,3) y = 4x-x2 (0,0) x = 3 X

8. Example 2 : • To Sketch and find the area bounded by the loop of the curve 3ay2 = x( x – a)2. • The curve is symmetric about the x – axis, • Therefore y is defined if x 0.

9. The curve intersects x – axis at (0,0) & (a, 0) • Therefore loop in formed between these point. • x = 0 i.e., y – axis is tangent to the curve at the origin. • The curve does not have any asymptotes.

10. Y (a,o) X 0 Graph of the given curve

11. The area bounded by the loop of the curve • = 2 area bounded by the portion of the loop of the curve, x – axis lying in the first quadrant

12. = X • Example 3: • To find the area inside the cardioide • r = a ( 1+ cos) and the circle r = 2 cos • Required area is given by XX

13. Area

14. Example 4: • To find the area bounded by the curve • y2(a-x) = x3 and its asymptote. • X = a is the asymptote to the curve. • The required area is given by • Area = 2 The area bounded by the curve and the asymptote lying in the first quadrant.

15. Y X o X=a

16. EXERCISES • 1. Find the area bounded by the one arch of the cycloid x = a ( - sin ) , y = a ( 1- cos ) and its base • 2. Find the area of the region lying above x – axis, included between the circle x2 + y2 = 2ax and the parabola y2 = ax. • 3. Find the area between the curve x ( x2 + y2) = a(x2 – y2) and its asymptote.

17. 4.Find the area bounded by the curve . • 5.Find the area bounded by the loops of the curve r2 = a sin  • 6. Find the area inside r = a ( 1 – cos ) and outside r = a sin  • 7: Find the area common to the curves r = a ( 1+ cos) and r = a ( 1 - cos)

18. 8. Find the area inside r = a and outside r = 2acos. • 9. Find the area bounded by the loops of the curve x3 + y3 = 3axy. • 10. Find the area bounded by the loops of the curve r = a sin3.

19. RECTIFICATION-LENGTH OF THE PLANE CURVE • The rectification is the process of determining the length of the arch of a plane curve. We know that the derivative of the arc of length of a curve is given by

21. The length of the arc of the curve is given by • Arc length s =

23. To find the length of the arc of the parabola x2 = 4ay measured from the vertex to one extremity of the latus rectum. • Here

24. The required arc length

25. To find the perimeter of the curve • The parameter equation of this curve is x = a cos3, y = a sin3.

26. Therefore Perimeter of the curve =6a

27. To find the perimeter of the curve • r = a ( 1+ cos).

29. EXERCISES • 1.Find the length of the arc of one arch of the cycloid x = a(+ sin), y = a ( 1- cos ). • 2.Find the length of the loop of the curve 3ay2 = x (x – a)2. • 3.Find the length of the arc of the curve of the centenary y = c cosh(x/c) measured from the vertex to any point (x ,y). • 4.Find the length of arc of the loop of the curve r r2 = a2 cos2.

30. VOLUME OF REVOLUTION Let a curve y = f(x) revolve about x–axis. Then the volume of the solid bounded by revolving the curve y = f(x), in between the lines x = a and x = b, about x – axis is given by

31. If the curve revolves about y – axis, the volume is given by • Examples : • 1.To find the volume of the solid obtained by revolving one arch of the curve x = a (+sin) , y = a (1 + cos) about its base

32. X- axis is the base of the curve • Therefore the required volume Y X a 0 -a o

34. 2. To find the volume bounded by revolving the curve y2(a – x ) = x3 about its asymptote. • X = a is the asymptote to the curve. Shifting the origin to the point (a, 0), we get the new coordinates X = x – a & Y = y – 0 = y. • Then the volume bounded by revolving the curve about the asymptote is given by

37. 2.To find the volume of the solid bounded by revolving the cardioide r = a(1+cos ) about the initial line • Required volume

39. PROBLEMS • The loop of the curve 3ay2=x(x – a)2 moves about the x – axis ; find the volume of the solid so generated. • Find the volume of the spindle shaped solid generated by revolving the curve about x – axis .

40. SURFACE AREA OF REVOLUTION The area of the surface of the solid obtained by revolving about x – axis, the arc of the curve y = f(x) intercepted between the points whose abscissa are a and b , is given by

41. Examples : • 1.To find the area of the surface of the solid generated by revolving on arch of the curve x= a (  – sin), y = a(1 – cos) about its base. • X =a( -sin ) ,y = a(1 - cos )

43. 2.To find the surface of the solid formed by revolving the curve r = a(1+cos) about the initial line ,

45. EXERCISES • 1.Find the area of the surface of the solid generated by revolving the arc of the parabola y2 = 4ax bounded by its latus rectum about x –axis. • 2.Find the area of the surface of revolution about the x – axis the curve x2/3 +y2/3 = a2/3 . • 3.Find the total area of surface of revolution of the curve r2 = a2cos2 about the initial line. • 4.Find the area of the surface of revolution of the loop of the curve 3ay2=x(x-a)2 about the x – axis.