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INTEGRAL CALCULUS, , Monday, October 18, 2021 9:26 PM, , , , , , , , , , , , , , , , Symbols/Terms/Phrases Meaning, * [f@ax Integral of f with respect to x, Fain [fade Integrand, xin [fade Variable of integration, Integrate Find the integral, ‘An integral of f A function F such that, F@) =f @), Integration The process of finding the integral, Constant of Integration ‘Any real number C, considered as, , , , constant function, , , , Z (sigma) mea, , » Any funenon :, , , , orl, ng summahon equivalent fo (, , d? fix) , = dependent voriabte, Xz (ndependent Vg viable, , , , Spat ay 7 ntenitely, “ genail change, , Zoi in %, , vaviable of Iategrane”, , Further, the following equation (statement) | f@) dx =E(Q@)+ 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., , , , pr et, , pest 43 a ee, , i i tanon, Integration] | Ditteven, , $0, we need to undestond thar, , p= d% is the Slope of foageat, , to the Curve, , gc: na a, that means, Ind ehinit¢ Mlegranoa ges, us a family of Curve whose slope of, , yruttc:, , Hovever, (ntegranon of pret gure, , tangent at o gen point “x! semains sant:

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1. The process of differentiation and integration are inverses of each other, , 2. The derivative of a function has a geometrical meaning, namely, the slope of the tangent to the, corresponding curve at a point., Similarly, the indefinite integral of a function represents geometrically:, a family of curves placed parallel to each other having parallel tangents at the points of, intersection of the curves of the family with the lines orthogonal (perpendicular) to the axis, representing the variable of integration., , NOTE : The derivative of a function, when it exists, is a unique function. The integral of a function is not, 50, However, they are unique upto an additive constant, i.e., any two integrals of a function differ by a, constant, , You are familiar with the notion of area. The formulae for areas of simple geometrical figures are, also known to you. For example, the area of a rectangle is length times breadth and that of a, triangle is half of the product of base and height. But how to deal with the problem of determination, of area of an irregular figure? The mathematical notion of integral is necessary in connection with, such problems., , Let us take a concrete example. Suppose a variable force f(x acts on a particle in its motion, along x - axis from x=a tox=b. The problem is to determine the work done (W) by the force on the, particle during the motion. This problem is discussed in detail in Chapter 6., , Figure 3.31 shows the variation of F(x) with x. If the force were constant, work would be simply, the area F(b-a) as shown in Fig. 3.31(i). But in the general case, force is varying ., , =, , , , , , F(X) FO), , , , , , , , , , , , Oa b of tH a *, (i) (ii), , To calculate the area under this curve [Fig. 3.31 (ii)], let us employ the following trick. Divide the, interval on x-axis from a to binto a large number (N) of small intervals: x,{=a) to x, X; 0X2; Xz to X;,, CURRENT Xy_1 toXy (=b). The area under the curve is thus divided into N strips. Each strip, is approximately a rectangle, since the variation of Fl over a strip is negligible. The area of the i””, strip shown [Fig. 3.31(ii)] is then approximately

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AA; = F(x) 0% - Xi-1) = FlxiAx, , where Axis the width of the strip which we have taken to be the same for all the strips. You may, wonder whether we should put F(%,,) or the mean of F(x) and F{x;.;) in the above expression. If we, take Nto be very very large (N--), it does not really matter, since then the strip will be so thin that, the difference between F(x) and F(x; ;) is vanishingly small. The total area under the curve then is:, , N N, , A=) AA, = DY Pear, , i=l i=l, , The limit of this sum as N-~ is known as the integral of F(x) over x from a to b. It is given a special, symbol as shown below:, , b, A=[Poidx = hia Z Fx) Ox, 4x70, , a, , The integral sign J looks like an elongated S, reminding us that it basically is the limit of the sum, , of an infinite number of terms., A most significant mathematical fact is that integration is, in a sense, an inverse of differentiation., , Suppose we have a function g (x) whose derivative is f(x), ie. f() = me), , The function g (x) is known as the indefinite integral of f(x) and is denoted as:, , ged = f foaax, , An integral with lower and upper limits is known as a definite integral. It is a number. Indefinite, integral has no limits; it is a function., , A fundamental theorem of mathematics states that +:, Gz upeer liar, , + B (init, J Sod de = gb = of0) - gta) t b= hover lt, , a, , As an example, suppose f(x) = x” and we wish to determine the value of the definite integral from, x=1tox=2. The function g (x) whose derivative is x“ is x°/3. Therefore,