Calculus
Cover All Concept’s in Differential Calculus including Asymptotes, Curvature ,Concavity, Convexity & Inflexion and Curve Tracing
Cover all concept’s In Differential Calculus-II including Partial Differentiation Approximate Calculations Maxima & Minima
Multiple Integrals
- The process of integration for one variable can be extended to the functions of more than one
variable. The generalization of definite integrals is known as Multiple Integral
Double Integrals
Consider the region R in the x, y plane we assume that R is a closed*, bounded** region in the x , y plane, by the curve y = f1(x), y = f2(x) and the lines x = a, x = b. Let us lay down a rectangular grid on R consisting of a finite number of lines
parallel to the coordinate axes. The N rectangles lying entirely within R (the shaded ones in Fig.4.1). Let (xr, yr) be an arbitrarily selected point inthe rth partition rectangle for each r = 1, 2, ..., N.Then denoting the area δxr · δyr = δSr
Thus, the total sum of areasSN = fx y r rrN ,=∑1δSrLet the maximum linear dimensions of eachportion of areas approach zero,and n increases indefinitely then the sum SN will approach a limit, “namely the double integral f x y dSR , and the value of this limit is given by
** (If you cant able to see Don't Worry, We will discuss it lateron.)
Beta & Gamma Functions
The first and second Eulerian Integrals which are also called “Beta and Gamma functions” respectively are defined as follows:
β (m, n) is read as “Beta m, n” and n is read as “Gamma n”. Here the quantities m and n are positive numbers which may or may not be integrals.
