Definite integral

We know how to find the area of a solid figure. For example, if we have a rectangle the product of length and width give its area. But what happens if we have to find the area under a curve y= f(x). The fundamental problem with the integral calculus is to find the area under a given curve. Now we see how to find the area under a given curve y=f(x) from a to b.

We divide the area under the curve into rectangles of equal width.

Take x₀= a and x_n= b where n is the number of rectangles. Then we find the area of each rectangle and find the limit of those areas.

∆x = (b-a)/n

S₁=∆x * f(x₁)

S₂=∆x * f(x₂)

…………………

…………….

S_n=∆x * f(x_n)

As the number of rectangles increases the width of each rectangle reduce

Then find the total sum S = S₁ + S₂ + …………….. + S_n

Therefore area under the curve = ∫_a ^b f(x) dx