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Definite Integrals

Integration as limit of sum

The summation notation or sigma notation enables us to write a sum with many (finitely or infinitely many) terms in the compact form. It uses the uppercase Greek letter Σ (sigma) to denote various kinds of sums.

For example, consider the sum: 2 + 4 + 6 + 8 + 10 + 12, in which each term is of the form '2i', where 'i' is one of the integer from 1 to 6. In sigma notation, this sum can be written as: , and read as "the summation of 2i, where i runs from 1 to 6".

In the notation , the Greek capital letter Σ stands for sum, the expression '2i' is called the summand, the numbers 1 and 6 are called the limits of summation [i.e., 1 is called the lower limit and 6 is called the upper limit of the summation], and 'i' is called the index of summation. The index of summation 'i' tells us that where to begin the sum (at the number below Σ) and where to end (at the number above Σ).

Rules for summation
Rule 1: The sum of the sums of two or more variables is equal to the sum of their summations, i.e., .
Rule 2: The difference of the sums of two or more variables is equal to the difference of their summations, i.e., .
Rule 3: The sum of a constant times the values of a variable is equal to the constant times the sum of the variable, i.e., , where 'p' is any constant.
Rule 4: The sum of a constant taken 'n' times is the constant times 'n', i.e., .
Summation formulae
If 'n' is any positive integer, then
. .
. .
Riemann Sums

Suppose that a function f(x) is continuous on a closed interval [a, b]. The graph of a function f(x) over a closed interval [a, b] is shown below, which may have positive values as well as negative values.

We then partition the interval [a, b] into 'n' sub-intervals by choosing n – 1 points, say x1, x2, x3, x4, x5, . . . . . . . . . . ., xn – 1, between a and b subject only to the condition that a = x0 < x1 < x2 < . . . . . . . . . . . < xn – 1 < xn = b. The set P = {x0, x1, x2, . . ., xn} is called a partition of the closed interval [a, b].

The partition P = {a = x0, x1, x2, x3, . . ., xn = b} divides the closed interval [a, b] into 'n' sub-intervals called [x0, x1], [x1, x2], [x2, x3], . . ., [xi – 1, xi], . . ., [xn – 1, xn] of lengths Δx1, Δx2, Δx3, . . ., Δxi, . . ., Δxn.

In each sub-interval, we select some number. Denote the number chosen from the ith sub-interval by ci. Then, on each sub-interval we stand a vertical rectangle that reaches from the x-axis to touch the curve at (ci, f(ci)). These rectangles could lie either above or below the x-axis.

On each sub-interval, we form the product f(ci) . Δxi. This product can be positive, negative or zero, depending on the value of f(ci). Finally, we take the sum of these products: Sn = f(ci)Δxi. This sum, which depends on the partition P and the choice of the numbers ci, is a Riemann sum for f(x) on the interval [a, b].

If we take the sample points to be right hand endpoints of the sub-intervals, then ci = xi and the Riemann sum = f(xi)Δxi. If we take the sample points to be left hand endpoints of the sub-intervals, then ci = xi – 1 and the Riemann sum = f(xi – 1)Δxi.

Definite Integrals

The definite integral as a limit of Riemann sums: Let f(x) be a function defined on a closed interval [a, b]. For any partition P of [a, b], let the numbers ci be chosen arbitrarily in the sub-intervals [xi – 1, xi]. If there exists a number 'I' such that = I, where "max . Δxi" is the length of the largest sub-interval in the partition P, then 'f(x)' is integrable on [a, b] and I is the definite integral of f(x) over [a, b].

The existence of definite integrals: All continuous functions are integrable. That is, if a function f(x) is continuous on a closed interval [a, b], then its definite integral over [a, b] exists.

The definite integral of a continuous function over [a, b]: Let f(x) be a continuous function defined on a closed interval [a, b] and let [a, b] be partitioned into 'n' sub-intervals of equal length . We let x0 (= a), x1, x2, . . . . . . . ., xn (= b) be the endpoints of these sub-intervals and we choose sample points c1, c2, . . . ., cn in these sub-intervals, so ci lies in the ith sub-interval [xi – 1, xi]. Then the definite integral of f(x) from a to b is: f(x) dx = .

The symbol was introduced by Leibniz and is called an integral sign. It is elongated 'S' and was chosen because an integral is a limit of sums. In the notation: f(x) dx, f(x) is called the integrand, 'a' is called the lower limit of integration, and 'b' is called the upper limit of integration. The symbol 'dx' indicates that x is the variable of integration.

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