INTEGRATION.
Consider that when the function y = f (x) is plotted, the graph of the function is a curve as shown in Fig. 5.01.
From the points P and Q,drop PL and QM perpendi - In c culars to the X-axis. Suppose that we are interested in finding the area PQML under the curve PQ. Since it is not a regular geometrical figure (such as a rectangle, a square, a circle, etc), no formula for evaluating the area is known to us. However, it is possible to calculate this area by making use of integral calculus. For this, we divide the whole area in to n elementary strips, say each of width Ax as shown in the figure. Let ABCD be the ith elementary strip. It follows that the area PQML is equal to the sum of the areas of these n strips i.e.
area PQML=\(
\sum\limits_{i = 1}^n {\text{area ABCD of the ith strip}}
\) ...............(5.01)
Suppose that the ith elementary strip corresponds to the points A (x, y) and
B (x +\(\Delta\) x, y +\(\Delta\) y). Since Ax is very small, the strip ABCD can be considered as a rectangle of height fore BC=AD = y and width \(\Delta\)x. Then, area of ith the elementary strip i.e.
area ABCD=yx \(\Delta\)x= f(x)\(\Delta\)x [y=f(x)]
Hence, from the equation (5.01), we have
area PQML=\(
\sum\limits_{i = 1}^n {f(x)\Delta x}
\)...........(5.02)
Now, if n is increased, the width \(\Delta x\) of an elementary strip will also decrease. In the limiting case, when \(
n \to \infty
\), \(
\Delta x \to 0
\). It follows that in that event, the above summation will be exactly equal to the area PQML. It is because, any error occuring in calculating the area PQML due to the approximation BCAD=y will get eliminated, as \(
\Delta x \to 0
\). Therefore,
area PQML= \(
\sum\limits_{i = 1}^n {f(x)\Delta x}
\)............(5.03)
If OL= a and OM= b, then in the language of calculus, the equation (5.03) is expressed as
area PQML = \(
\int\limits_a^b {f(x)\Delta x}
\)
and is read as integral of f (x) w.r.t. x between the limits x= a and x = b.
Basically, integration is summation. The symbol for integration owes its origin to letter S (for summation). The letter S, when stretched, takes the shape of
symbol
The concept of integration is used in physics to make measurements, when a physical quantity varies in a continuous manner. The following are a few
examples:
1. Work done by a force is given as the product of the force (F) applied on the body and the distance (x) moved i.e. W = \(
F \times x
\)
The above expression holds only, when the force applied on the body remains constant over the whole distance. In case, the force does not remain constant over the whole distance, work done is calculated by making use of the concept of integration. The force F can be assumed to remain constant over an infinitesimally small distance dx. Then, small amount of work done,
dW = F dx
and the work done to move the body through the distance x,
W= \(
\int\limits_0^x {dW}
\)
or W= \(
\int\limits_0^x {Fdx}
\)
2. Impulse due to a force is given as the product of the force (F) and the time (t) for which it acts, provided the force remains constant during the whole time interval. Again, in the situation, when the force does not remain constant during the whole time interval (such as, when a cricketer hits the ball with his bat), the impulse due to the force can not be calculated. It is calculated by making use of the cocept of integration as explained above for calculating the work done. As such, the impulse due to the force is given by
I= \(
\int\limits_0^t {Fdt}
\)
INTEGRATION AS INVERSE OF DIFFERENTIATION
Integration is the process of finding the function, whose derivative is given. For this reason, the process of integration is called inverse process of differentiation
Consider a function f (x), whose derivative w.r.t. x is another function \(
f^1 (x)
\) i.e.
\(
\frac{d}
{{dx}}[f(x)] = f^1 (x)
\)
If differentiation of f(x) w.r.t. x is equal to \(
f^1 (x)
\), then f(x)+c is called the integration of \(
f^1 (x)
\), where c is called constant of integration*.
Symbolically, it is written as
\(
\int {f^1 (x)dx = f(x) + c}
\)
Here, \(
f^1 (x)
\) is called integrand, \(
f^1 (x)
\)dx is called element of integration and the symbol is the sign for integration.
Let us proceed to obtain integral of \(x^n\) w.r.t. x. We know that
\(
\frac{d}
{{dx}}(x^{n + 1} ) = (n + 1)x^n
\)
Since the process of integration is the inverse of differentiation,
\(
\int {(n + 1)x^n dx = } (x^{n + 1} )
\) or \(
(n + 1)\int {x^n dx = } (x^{n + 1} )
\)
or \(
\int {x^n dx = } \frac{{(x^{n + 1} )}}
{{n + 1}}
\)
The above formula holds for all values of n, except n = 1. It is because, for n = -1,
\(
\int {x^n dx = } \int {x^{ - 1} } dx = \int {\frac{1}
{x}dx}
\)
Since \(\frac {1}
{x}\) is differential coefficient of \(
\log _e x
\) i.e.
\(
\frac{d}
{{dx}}\log _e x = \frac{1}
{x}
\) ,
\(
\int {\frac{1}
{x}dx = } \log _e x
\)
Similarly, the formulae for integration of some other functions can be obtained from our knowledge of the differential coefficients of various functions.
BASIC FORMULAE FOR INTEGRATION
1.\(
\int {x^n dx = \frac{{(x^{n + 1} )}}
{{n + 1}}}
\)+c, provided \(
n \ne 1
\)
2. \(
\int {\sin xdx} = - \cos x + c
\) \(
\left( {\frac{d}
{{dx}}\left( {\cos x} \right) = - \sin x} \right)
\)
3. \(
\int {\cos xdx} = \sin x + c
\) \(
\left( {\frac{d}
{{dx}}\left( {\sin x} \right) = \cos x} \right)
\)
4. \(
\int {\frac{1}
{x}dx = } \log _e x + c
\) \(
\left( {\therefore \frac{d}
{{dx}}\left( {\log _e x} \right) = \frac{1}
{x}} \right)
\)
5. \(
\int {e^x dx} = e^x + c
\) \(
\left( {\therefore \frac{d}
{{dx}}\left( {e^x } \right) = e^x } \right)
\)
THEOREMS OF INTEGRATION
Theorem 1. The integral of the product of a constant and a function of x is equal to the product of the constant and integral of that function. Mathematically,
\(
\int {\operatorname{c} udx} = c\int {udx}
\)
where c is a constant and u is a function of x.
Theorem 2. The integral of the sum (or difference) of a number of functions is equal to the sum (or difference) of their integrals. of Mathematically,
\(
\int {\text{(u \pm v \pm w) dx = }\int {\text{u dx}} \text{ \pm }\int {\text{v dx}} \text{ \pm }\int {\text{w dx,}} }
\)
where u, u and w are functions of x
DEFINITE INTEGRALS
When a function is integrated between two specified limits, called lower and upper limits, it is called a definite integral.
If \(
\frac{d}
{{dx}}f\left( x \right) = f^1 (x)
\)
then \(
\int {f^1 (x)}
\) dx is called an indefinite integral
and \(
\int\limits_a^b {f^1 (x)}
\) dx is called a definite integral.
Here, a and b are called lower and upper limits of the variable x.
After carrying out integration, the result is evaluated between upper and lower limits as explained below:
\(
\int\limits_a^b {f^1 (x)} = |f\left( x \right)|_a^b = f(b) - f(a)
\)