Numerical Integration Class 11 Mathematics Solutions | Exercise - 20.1

  Chapter - 20.1  
Numerical Integration

We all have to study numerical integration in class 11 mathematics. It is somehow similar to numerical computations chapter. But being similar doesn't mean we will be able to solve each and every questions given in the exercise. We all may get stuck somewhere while solving the questions. 

Inorder to help you out we have brought the complete solution PDF of class 11 numerical integration chapter. We can only provide you the solutions of first exercise because we don't have the solutions of second exercise. Before we move towards the solution section, let's have a quick look at the topics we have to study in this chapter.

Exercise - 20.1

Introduction

Numerical implies something that is associated with numbers; and integration is finding a sum or total value. Numerical integration thus literally means sum total in tems of numbers.

In mathematics, numerical integration is the name given to numerical computation of integral when it becomes very difficult or infeasible or impossible to find its exact value. In other words, numerical integration is concerned with the computation of an approximate value for the definite integral rather than finding an exact value. Various methods of numerical are known. We shall however be concerned with two methods, namely, the Trapezoidal Rule and Simpson's Rule.

Definite Integral and Area Computation

The fundamental Theorem of Calculus gives us a method of computing the definite integral of a given continuous function f from a to b. The method is to find, by trial and error, an anti-derivative F of f and then to use the equation;

When the method works, it provides an exact value for the inegral. However, the method succeeds only if the anti-derivative happens to be a function that can be described in a simple way. For many integrals one cannot find a formula for the anti-derivative, and the method fails.

Such integrals can still be computed approximately using numerical intgration. A computation of this kind becomes possible once we agree that a definite integral is nothing but a limit of sum of rectangles or trapeziums constructed to cover the region bounded by the graph of the function f, two vertical lines x a and x = b and the x-axis

Composite Trapezoidal Rule

One way of arriving at a more accurate numerical approximation of a definite integral is to break up the given interval [a, b] into some number n of sub-intervals, imply elementary trapezoidal rule to each interval, and then add up all the results. Use of trapezium gives rise to what is known as the trapezoidal rule. Here is how trapezoidal rule goes: