Integral calculus study notes and practice questions., Study notes of Mathematics

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INTRODUCTION x If fis continuous, then jf f(t) dt is an antiderivative of f. and a b says that f f(x) dx can be found by evaluating F(b) — F(a), a where F is an antiderivative of f. We need a convenient notation for antiderivatives that makes them easy to work with. Because of the relation given by the Fundamental Theorem between antiderivatives and integrals, the notation frooax is traditionally used for antiderivative of f and is called and indefinite integral. Thus Jecoax = F(x) + C means F (x)= fx) For example, we can write 3 d S Jx?ax =*_4¢ because — [+c] =x? 3 dx | 3 So we can regard an indefinite integral as representing an entire family of function (one antiderivative for each value of the constant C). You should distinguish carefully between definite and b indefinite integrals. A definite integral [ f(x) dx is anumber, a whereas an indefinite integral fe (x)dx is family of functions. The connection between them is given by part 2 of the Fundamental Theorem. If fis continuous on [a, b] then [ f(x) dx= Jreoax? The effectiveness of the Fundamental Theorem depends on having a supply of antiderivatives of functions. We adopt the convention that when a formula for a general indefinite integral is given, itis valid only on an interval. Thus, we write | 1. Ja Xx =-—+C Ms x With the understanding that itis valid on the interval (0, %) or on the interval (-, 0). This true despite the faet that the antiderivative of the function Ax) = U/x2.x #0, is general auc FO)=} -—+C, if x>0 x if x<0 Note : If ,(x) & $,(x) are two antiderivatives ofa function f(x) on [a, b] the difference between them is constant GEOMETRICAL INTERPRETATION OF INDEFINITE INTEGRATION Je@xdax = F(x) +C =y(say), represents a family of curves, The different values of C will correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself. This is the geometrical interpretation of indefinite integral. This is the geometrical interpretation of indefinite integral. Let f(x) = 2x. Then fr@ax =x°+C. For different values of C, we get different integral. But these integrals are very similar geometrically. Thus, y= x? +C, where C is arbitrary constant, represents a family of integrals. By assigning different valeus to C, we get different members of the family. These together constitute the indefinite integral. In this case, each integral represents a parabola with its axis along y-axis. Ifthe line x =a intersects the parabolas y= x”, y= x" + I, y=x?+2,y=x?-1, y=x?-2 at P,, P,, P, P,P, ete, d Os AaB tespectively, then a at these point equals 2a. This indicates theat the tangents to the curves at these points are parallel. ;