Approximating the Area Under a Curve

Let y = f(x) be a continuous function defined on the closed interval [a, b]. To find the area of the region that lies under the curve y = f(x) from a to b, where f(x) ≥ 0 for all 'x' in [a, b], divide the interval into 'n' strips of equal width Δx = . These strips divide the interval [a, b] into 'n' sub-intervals, called, [a = x_{0}, x_{1}], [x_{1}, x_{2}], . . ., [x_{n – 1}, x_{n} = b].

In each sub-interval we select some number. Denote the number chosen from the i^{th} sub-interval by c_{i}. Construct an approximating rectangle above the sub-interval [x_{i – 1}, x_{i}] with the height f(c_{i}) and the width Δx. Then the area of this rectangle is: f(c_{i})Δx.

What we think of intuitively as the area of the region that lies under the curve y = f(x) from a to b is approximated by the sum of the areas of these rectangles, which is: f(c_{1})Δx + f(c_{2})Δx + . . . . + f(c_{n})Δx. This approximation appears to become better and better as the number of strips increases, that is, as n → ∞. Therefore, we define the area of the region that lies under the curve y = f(x) from a to b in the following way.

**Definition: **The area of the region that lies under the graph of the continuous function 'f' is the limit of the sum of the areas of approximating rectangles: [f(c_{1})Δx + f(c_{2})Δx + . . . . + f(c_{i})Δx + . . . . + f(c_{n})Δx], where c_{i} is any point in the i^{th} sub-interval [x_{i – 1}, x_{i}].

**Left endpoint approximation: **If c_{i} is chosen to be the left endpoint of the sub-interval [x_{i – 1}, x_{i}], then c_{i} = x_{i – 1} and we have: L_{n} = [f(x_{0}) + f(x_{1}) + . . . . . . . . . . . . + f(x_{n – 1})]Δx = .

**Right endpoint approximation: **If c_{i} is chosen to be the right endpoint of the interval [x_{i – 1}, x_{i}], then c_{i} = x_{i} and we have: R_{n} = [f(x_{1}) + f(x_{2}) + . . + f(x_{n})]Δx = .

**Midpoint rule: ** If c_{i} is chosen to be the midpoint of the interval [x_{i – 1}, x_{i}], then c_{i} = x_{i} = [x_{i – 1} + x_{i}] and we have: M_{n} = [f(x_{1}) + f(x_{2}) + . . . . . . . . . . + f(x_{n})]Δx = .

**Trapezoidal rule: **This is another method for approximating the area under a curve, which results from averaging the left endpoint and the right endpoint approximations. Therefore, according to trapezoidal rule, the area of the region that lies under the curve y = f(x) from a to b is: = = .