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30 5 INTEGRATION Part IV Lecture 4 5 Integration 5.1 Dierentials Let us rst understand what do we mean by dierentials Dierential of x is ∆x and is denoted as dx lim ∆x→0 So a dierential dx is a small dynamic change in x (Remember dierential of x is d(x) = dx) Example. See the following worked examples of dierentials 1. Dierential of y is dy 2. Dierential of f(x) is d(f(x)) = f ′ (x)dx 3. Dierential of sin x is d(sin x) = cos x dx 4. Dierential of sin x 2 is d(sin x 2 ) = cos x 2 2x dx 5. Dierential of f(g(x)) is d(f(g(x))) = f ′ (g(x)) g ′ (x)dx This can be seen in a dierent way, dierential of f(x) is d(f(x)) this can be produced by dierenting f(x) with respect to the independent variable here i.e. x d df(x) f(x) = dx dx = f ′ (x) ⇒ d(f(x)) = f ′ (x) dx 5.2 Indenite Integration Denition. It is dened as reverse operation of Dierentiation. Anti-derivative. Also called Notation: ´ f(x)dx : read as integral of f(x) dx which means we are trying to nd that function whose derivative with respect to x is f(x) for example: ´ sin x dx = − cos x : means derivative of − cos x is sin x 1. 2. 3. 4. d dx (xn+1 ) = (n + 1) · x n ↔ ´ x n dx = xn+1 n + 1 d dx (ax ) = a x · log e a ↔ ´ a x dx = ax log e a d dx (log e x) = 1 x ↔´ 1 x dx = log e x d dx (ex ) = e x ↔ ´ e x dx = e x 5. Trigonometric functions where n ≠ −1
5.3 Denite Integration 31 (a) d dx (sin x) = cos x ↔ ´ cos x dx = sin x (b) d dx (cos x) = − sin x ↔ ´ sin x dx = − cos x (c) d dx (tan x) = sec2 x ↔ ´ sec 2 x dx = tan x 5.3 Denite Integration Denition. It is the area under the curve y = f(x) from x varying from a to b (see the diagram) Notation : x=b ´ f(x) : read as integral of f(x) from x = a to x = b. x=a Example. We see here if we have a function y = f(x) and want to nd the area bounded between x = a (point A) and x = b (point B) along x-axis. And along y-axis its bounded by y = f(x) and x-axis. This is represented by denite integral ´b a f(x) dx We can nd this area under the curve (as its called) using approximation. We cover the area with rectangles and keep on increasing the number of such rectangles (just as we worked out to approximate the area of circle with area of n-gon). And then we nd limit of this process which turns out to be denite integral stated above. In the gure below, we have shown some stages of increasing the number of rectangles between x = a & x = b. Here n = 4, 7, 12, 26, 60, the limits case nally. 5.3.1 Way to evaluate Denite Integrals This is called the Fundamental Law
Page 1 and 2: Foundation of Calculus July 15, 200
Page 3 and 4: CONTENTS 3 5.4 Methods of Solving I
Page 5 and 6: 1.2 Dene slope for two points 5 1.2
Page 7 and 8: 1.4 Slopes of line 7 1.4 Slopes of
Page 9 and 10: 2.4 Domain and range of a function
Page 11 and 12: 11 Part II Lecture 2 3 Examples of
Page 13 and 14: 3.3 Rotating a marble tied to a cor
Page 15 and 16: 3.5 Denition of Limit 15 3.5 Deniti
Page 17 and 18: 3.7 Worked out problems in limits 1
Page 19 and 20: 19 (b) lim x→1 + (c) lim x→1
Page 21 and 22: 21 dy dx = lim f(x + h) − f(x) h
Page 23 and 24: 4.1 Derivatives of other functions
Page 25 and 26: 4.3 Rate of change 25 Obviously, if
Page 27 and 28: 4.6 Chain Rule and friends 27 Simil
Page 29: 4.8 Derivative of a composite funct
Page 33 and 34: 5.3 Denite Integration 33 from the
Page 35 and 36: 5.4 Methods of Solving Integrals 35
Page 37: REFERENCES 37 References [1] Thomas

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