Class 12 Maths Integrals Definite Integral

Definite Integral

A definite integral is denoted by aʃb f(x) dx, where a is called the lower limit of the integral and b is called the upper limit of the integral. The definite integral is introduced either as the limit of a sum or if it has an anti derivative F in the interval [a, b], then its value is the difference between the values of F at the end points, i.e., F(b) - F(a).

 Class_12_Maths_Integrals_Definite_Integration

Definite integral as the limit of a sum:

The definite integral aʃb f(x) dx is the area bounded by the curve y = f (x), the ordinates x = a, x = b and the x-axis. To find this area Let us consider the region PQRSP between this curve and the ordinates x = a and x = b as shown in the given figure.

Now, divide the interval [a, b] into n equal sub intervals denoted by [x0, x1], [x1, x2], ……..,[xr-1, xr], ………..

 [xn-1, xn] where x0 = a, x1 = a + h, x2 = a + 2h, ……….,xr = a + rh, xn = b = a + nh or h = (b - a)/n

 Again as n -> ∞, h -> 0

 The region PRSQP under consideration is the sum of n sub regions, where each sub region is defined on sub intervals [xr – 1, xr], r = 1, 2, 3, …, n.

Now, to find the area of the region PQRSQ is calculated as

Limx->∞ Sn = area of the region PQRSQ = aʃb f(x) dx

   = limh->0 h[f(a) + f(a + h) + ………….+ f{a + (n - 1)h}]

   = (b - a) * limh->∞ (1/n)[f(a) + f(a + h) + ………….+ f{a + (n - 1)h}]

 So, aʃb f(x) dx = (b - a) * limh->∞ (1/n)[f(a) + f(a + h) + ………….+ f{a + (n - 1)h}]

 Where h = (b - a)/n as n -> ∞

 This is known as the definition of definite integral as the limit of sum.

Problem: Evaluate the definite integral as limit of sums

ʃ05 (x + 1) dx

Solution:

We know that

ʃab f(x) dx = (b - a)limn->∞ (1/n)[f(a) + f(a + h) + …………..+ f{a + (n - 1)h}], where h = (b - a)/n

Here a = 0, b = 5 and f(x) = x + 1

So, h = (5 - 0)/n = 5/n

Now, ʃ05 (x + 1) dx = (5 - 0)* limn->∞ (1/n)[f(0) + f(5/h) + …………..+ f{5(n - 1)/n}]

                                 = 5 * limn->∞ (1/n)[1 + (5/n + 1) + …………..+ {1 + 5(n - 1)/n}]

                                 = 5 * limn->∞ (1/n)[(1 + 1 + 1 + …….n times) + {5/n + 2 * 5/n + …………..+ 5(n - 1)/n}]

                                 = 5 * limn->∞ (1/n)[(1 + 1 + …….n times) + (5/n){1 + 2 + 3 +…………..+ (n - 1)}]

                                 = 5 * limn->∞ (1/n)[n + (5/n) * n(n - 1)/2]

                                 = 5 * limn->∞ (1/n)[n + 5(n - 1)/2]

                                 = 5 * limn->∞ (n/n)[1 + 5(1 – 1/n)/2]

                                 = 5 * [1 + 5(1 – 1/∞)/2]

                                 = 5 * [1 + 5(1 – 0)/2]

                                 = 5 * [1 + 5/2]

                                 = 5 * (7/2)

                                 = 35/2

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