Question

Definite integral as the limit of a sum

Answer 1

Let f be a continuous function defined in close interval [a, b]. Assume that all the values taken by the function are non-negative, so the graph of the function is a curve above the x-axis.

The definite integral \(\int^b_a\) f(x)dx is the area bounded by the curve y = f(x), the ordinates x = a, x = b and the x-axis. To evaluate this area, consider the region PRSQP between this curve, x-axis and ordinates x = a and x = b.

Divide the interval [a, b] into n equal sub intervals denoted by [x₀, x₁], [x₁; x₂], ............... [x_r-1, x_r], ............ [x_n-1, x_n] n. Let width of each sub - interval is h where x₀ = a, x₁ = a + h, x₂ = a + 2h ..........., x_r = a + rh and x_n = b = a + nh ⇒ n = \(\frac{b-a}{h}\)

We note that as n → ∞ h → ∞

The region PRSQP is the sum of n sub-regions.

Clearly, area of rectangle (MNCD) < area of region (MNBDM) < area of rectangle (MNBA). ...(1)

If h → 0 all the three areas shown in (1) become nearly equal to each other.

If S_n and S_n' denote the sum of areas of all lower rectangle and upper rectangles raised over sub-intervals [x_r-1, x_r] for r = 1,2,....... n respectively, then

Here, at a = x0 length of rectangle is f(x₀) and breadth of rectangle is h.

∴ Area of lower rectangle = f(x₀) × h = hf(x₀)

And at point x₁ length of rectangle is f(x₂) and breadth of rectangle is h.

Area of upper rectangle = f(x₁) ≠ h = hf(x₁)

For any arbitrary sub-interval [x_r-1 x_r], we have S_n < (MNBDM) area of region (MNBDM) < S_n' ...(4)

As n → ∞ strips become narrower and narrower. It is assumed that the limiting values of S_n and S_n' are same in both cases and the common limiting value is the required area under the curve.

Symbolically, we write

The above expression (6) is known as the definition of definite integral as the limit of sum.

Some useful results for direct applications :

(i) Sum of n natural numbers

Σn = 1 + 2 + 3+ ... (n-1) + n = \(\frac{n(n+1)}{2}\)

(ii) Sum of square of n natural numbers

Σn² = 1² + 2² + 3² + ... n² = \(\frac{1}{6}\) n(n +1) (2n + 1)

(iii) Sum of cube of n natural numbers

Σn³ = 1³ + 2³ + 3³ + ... n³ = \([\frac{n(n+1)}{2}]^2\)

(iv) Sum of n terms of A.P.

a + (a + d) + (a + 2d) +... a + (n - 1 )d = \(\frac{n}{2}\) [2a + (n - 1)d]

(v) Sum of n terms of G.P.

a + ar + ar² + ... + ar^n-1 = \(\frac{a(r^n-1)}{(r-1)}\)  when r > 1

(vi) sin a + sin (a + h) + sin (a + 2h) +....+ sin [a + (n - 1 )h]

(vii) cos a + cos (a + h) + cos (a + 2h) +...+ cos [(a + (n - 1 )h]

Remark :

The value of the definite integral of a function over any particular interval depends on the function and the interval, but not on the variable of integration that we choose to represent the independent variable. If the independent variable is denoted by t or u instead of x, we simply write the integral as \(\int^b_a\) f(t)dt or \(\int^b_a\) f(u)du instead of \(\int^b_a\) f(x)dx. Hence, the variable of integration is called a dummy variable.