Category: 4. Integrals

The primitive value of the function found by the process of integration is called an integral. An integral is a mathematical object that can be interpreted as an area or a generalization of area. When a polynomial function is integrated the degree of the integral increases by 1.

Using integration, we can find the distance given the velocity. Definite integrals form the powerful tool to find the area under simple curves, the area bounded by a curve and a line, the area between two curves, the volume of the solids. The displacement and motion problems also find their applications of integrals. The area of the… Continue reading Applications of Integral Calculus

There are several methods adopted for finding the indefinite integrals. The prominent methods are: Finding integrals by integration by substitution method Finding integrals by integration by parts Finding integrals by integration by partial fractions. Finding Integrals by Substitution Method A few integrals are found by the substitution method. If u is a function of x, then u’ = du/dx.… Continue reading Methods to Find Integrals

We can remember the formulas of derivatives of some important functions. Here are the corresponding integrals of these functions that are remembered as standard formulas of integrals. ∫ xn dx=xn+1 /n+1+C, where n ≠ -1 ∫ dx =x+C ∫ cosxdx = sinx+C ∫ sinx dx = -cosx+C ∫ sec2x dx = tanx+C ∫ cosec2x dx = -cotx+C ∫ sec2x dx… Continue reading Integrals Formulas

Let us study the properties of indefinite integrals to work on them.  The derivative of an integral is the integrand itself. ∫ f(x) dx = f(x) +C Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. ∫ [ f(x) dx -g(x) dx] =0 The integral of the… Continue reading Properties of Integral Calculus

Integral calculus is used for solving the problems of the following types. a) the problem of finding a function if its derivative is given. b) the problem of finding the area bounded by the graph of a function under given conditions. Thus the Integral calculus is divided into two types. Definite Integrals (the value of… Continue reading Types of Integrals

We define integrals as the function of the area bounded by the curve y = f(x), a ≤ x ≤ b, the x-axis, and the ordinates x = a and x =b, where b>a. Let x be a given point in [a,b]. Then  b∫a f(x)dx ∫abf(x)dx represents the area function. This concept of area function leads… Continue reading Fundamental Theorems of Integral Calculus

F(x) is called an antiderivative or Newton-Leibnitz integral or primitive of a function f(x) on an interval I. F'(x) = f(x), for every value of x in I. Integral is the representation of the area of a region under a curve. We approximate the actual value of an integral by drawing rectangles. A definite integral of a… Continue reading Definition of Integral

Integrals are the values of the function found by the process of integration. The process of getting f(x) from f'(x) is called integration. Integrals assign numbers to functions in a way that describe displacement and motion problems, area and volume problems, and so on that arise by combining all the small data. Given the derivative f’ of the function… Continue reading What is Integral Calculus?

Integral calculus helps in finding the anti-derivatives of a function. These anti-derivatives are also called the integrals of the function. The process of finding the anti-derivative of a function is called integration. The inverse process of finding derivatives is finding the integrals. The integral of a function represents a family of curves. Finding both derivatives and integrals form… Continue reading Integral Calculus