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- 1. DEFINITE INTEGRATION
- 2. DEFINITIONLet f(x) be a continuous function defined on [a, b], b f ( x ) dx = F(x) + c. Then f ( x ) dx = F(b) – F(a) is called a definite integral. This formula is known as Newton- Leibnitz formula.Note : The indefinite integral is f ( x ) dx a function of x, where as b definite integral f ( xisdx ) a number. a b b Given f ( x ) dx we can find f ( x ) dx, but given we f ( x ) dx a a cannot find f ( x.) dx
- 3. EXAMPLES
- 4. PROPERTIES OF DEFINITE INTEGRAL
- 5. b b1. f ( x) dx f (t ) dt a a i.e. definite integral is independent of variable of integration. b a2. f ( x) dx f ( x) dx a b b c b3. f ( x) dx f ( x) dx f ( x) dx a a c where c may lie inside or outside the interval [a, b].
- 6. EXAMPLES
- 7. a a4. f ( x) dx ( f ( x) f ( x)) dx a 0 a = 2 f ( x) dx , if f(–x) = f(x) i.e. f(x) is even. 0 = 0 , if f(–x) = –f (x) i.e. f(x) is odd. b b5. f ( x) dx f (a b x) dx a a a a Further f ( x) dx f (a x) dx 0 0
- 8. EXAMPLES
- 9. EXAMPLES
- 10. 2a a6. f ( x) dx ( f ( x) f (2a x)) dx 0 0 a = 2 f ( x) dx , if f(2a – x) = f(x) 0 = 0 , if f(2a – x) = – f(x)
- 11. EXAMPLES
- 12. 7. PERIODIC FUNCTIONIf f(x) is a periodic function with period T, then nT Ti. f ( x) dx n f ( x) dx , n Z 0 0 a nT Tii. f ( x) dx n f ( x) dx , n Z , a R a 0 nT Tiii. f ( x) dx (n m) f ( x) dx , m,n Z mT 0 a nT aiv. f ( x) dx f ( x) dx , n Z , a R nT 0 b nT av. f ( x) dx f ( x) dx , n Z , a, b R a nT 0
- 13. EXAMPLES
- 14. 8. SANDWICH THEOREMIf (x) f(x) (x) for a x b, then b b b ψ(x) dx f (x) dx (x) dx a a a
- 15. 9. If m f(x) M for a x b, then b m(b a) f ( x) dx M (b a) aFurther if f(x) is monotonically decreasing in (a, b) then b f (b)(b a) f ( x) dx f (a)(b a) aand if f(x) is monotonically increasing in (a, b) then b f (a)(b a) f ( x) dx f (b)(b a) a
- 16. b b10. f ( x) dx f ( x) dx a a b11. If f(x) 0 on [a, b] then f ( x) dx 0 a
- 17. EXAMPLES
- 18. LEIBNITZ THEOREM h( x)If F ( x) f (t ) dt , then g ( x) dF ( x) h ( x) f (h( x)) g ( x) f ( g ( x)) dxProof : Let P(t ) f (t ) dt h( x) F ( x) f (t ) dt P(h( x)) P( g ( x)) g ( x) dF ( x) P (h( x))h ( x) P ( g ( x)) g ( x) dx f ( h( x)) h ( x) f ( g ( x)) g ( x)
- 19. EXAMPLES
- 20. DEFINITE INTEGRAL AS A LIMIT OF SUMLet f(x) be a continuous real valued function defined on the closed interval [a, b] which is divided into n parts as shown in figure. y = f(x) a a+h a+2h ................ a+(n-1)h a+nh=b xThe point of division on x-axis area, a + h, a + 2h ..........a + (n – 1)h, a + nh ,where b a = h. n
- 21. Let Sn denotes the area of these n rectangles. Then, Sn = hf(a) + hf(a + h) + hf(a + 2h) + ........+hf(a + (n – 1)h) Clearly, Sn is area very close to the area of the region bounded by curve y = f(x), x–axis and the ordinates x = a, x = b. b Hence f ( x) dx Lim Sn n ab n 1 n 1 b a (b a)r f ( x) dx Lim h f (a rh) Lim f aa n r 0 n r 0 n n
- 22. Note :1. We can also write Sn = hf(a + h) + hf (a + 2h) + .........+ hf(a + nh) and b n b a b a f ( x) dx Lim f a r a n r 0 n n2. If a = 0, b = 1, 1 n 1 1 r f ( x) dx Lim f 0 n n n 0 r
- 23. Steps to express the limit of sum as definite integrali. Replace by x, by dx and by r 1 Lt n n n Evaluateii. r by putting least and greatest Lim values of r n lower and upper limits respectively. as nFor example, pn p 1 r Lim f f ( x) dx 1 n n n r 0 r r ( Lim 0, Lim p) n n r 1 n n r np
- 24. EXAMPLES
- 25. REDUCTION FORMULAE IN DEFINITE INTEGRALS 2 n n 1 If In sin x dx , then In In 2 0 nNote : 2 2 i. sin n x dx cos n x dx 0 0 n 1 n 3 n 5 ii. In ........I 0 or I1 n n 2 n 4 according as n is even or odd. I0 = /2 , I1 = 1. n 1 n 3 n 5 1 ........ . if n is even n n 2 n 4 2 2 Hence , In = n 1 n 3 n 5 2 ........ . 1 if n is odd n n 2 n 4 3
- 26. 4 If I n tan n x dx , then 0 1 In In 2 n 1 2 If I m,n sin m x cos n x dx, then 0 m 1 I m,n Im 2, n m n
- 27. Note : m 1 m 3 m 51. I m,n .......I 0, n or I1, n m n m n 2 m n 4 according as m is even or odd. 2 I 0,n cos n x dx and 0 2 n 1 I1,n sin x cos x dx 0 n 1
- 28. WALLI’S FORMULA (n 1) (n 3) (m 5) .........( n 1) (n 3) (n 5)....... w hen both m, n are even (m n) (m n 2) (m n 4)........ 2Im,n= (m 1) (m 3) (m 5) .........( n 1) (n 3) (n 5)....... otherw ise (m n) (m n 2) (m n 4)........
- 29. EXAMPLES
- 30. AREA
- 31. CURVE TRACINGTo find the approximate shape of a curve, the following procedure is adopted in order: Symmetry : i. Symmetry about x - axis: If all the powers of y in the equation are even then the curve is symmetrical about the x - axis. E.g.: y2 = 4 a x.
- 32. ii. Symmetry about y – axis : If all the powers of x in the equation are even then the curve is symmetrical about the y - axis. E.g.: x2 = 4 a y.iii. Symmetry about both axis : If all the powers of x and y in the equation are even, the curve is symmetrical about the axis of x as well as y . E.g.: x2 + y2 = a2.
- 33. iv. Symmetry about the line y = x : If the equation of the curve remains unchanged on interchanging x and y , then the curve is symmetrical about the line y = x. E.g.: x3 + y3 = 3 a x y.v. Symmetry in opposite quadrants: If the equation of the curve remains unaltered when x and y are replaced by - x and - y respectively, then there is symmetry in opposite quadrants. E.g.: x y = c2.
- 34. Find the points where the curve crosses the x-axis and also the y-axis. Find dy/dx and equate it to zero to find the points on the curve where you have horizontal tangents. Examine if possible the intervals when f (x) is increasing or decreasing. Examine what happens to ‘y’ when x or x - .
- 35. Asymptotes : Asymptote(s) is(are) line(s) whose distance from the curve tends to zero as point on curve moves towards infinity along branch of curve. i. If f(x) = or f(x) = – , then x = a is asymptote of y = f(x). Lt Lt x a x a ii. If f(x) = k or f(x) = k , then y = k is asymptote of y = f(x). Lt Lt x x iii. If = m1 , (f(x) – m1x) = c1 , then y = m1x + c1 is an asymptote. Lt f ( x ) Lt x x (inclined to right) x iv. If = m2 , (f(x) – m2x) = c2 , then y = m2x + c2 is an Lt f ( x ) Lt x x x asymptote (inclined to left)
- 36. EXAMPLES Find asymptote of y = e–x Ans. y = 0 is asymptote Find asymptotes of xy = 1 and draw graph. Ans. x = 0 is asymptote & y = 0 is asymptote
- 37. QUADRATURE If f(x) 0 for x [a, b], then area bounded by curve y b = f(x), x-axis, x = a and x = b is . f ( x ) dx a If f(x) 0 for x [a, b], then area bounded by curve y b = f(x), x-axis, x = a and x = b is – f ( x ) .dx a
- 38. Note : If y = f(x) does not change sign an [a, b], then area bounded by y = f(x), x-axis between ordinates x = a, x b = b is . f ( x ) dx a If f(x) > 0 for x [a,c] and f(x) < 0 for x [c,b] (a < c < b) then area bounded by curve y = f(x) and x–axis between c b x = a and x = b is . f ( x ) dx f ( x ) dx a c
- 39. If f(x) > g(x) for x [a,b] then area bounded by curves y = f(x) and y = g(x) between ordinates x = a and x = b is b . f ( x ) g( x ) dx a If g (y) > 0 for y [c,d] then area bounded by curve x = g(y) and y–axis between abscissa y = c and d y = d is g( y )dy . y c
- 40. EXAMPLES
- 41. EXAMPLES

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