Physics

Mathematics used in Physics

Area Calculation using Limits

Class 11

⚡ Quick Summary

We can find the area under a curve by dividing it into tiny rectangles, summing their areas, and then making the rectangles infinitely small. This is called integration.

I = lim (Δx→0) Σ f(xi)Δx = ∫ab f(x) dx

The area under a curve y = f(x) from x = a to x = b can be approximated by dividing the area into N rectangles of width Δx.
The area of each rectangle is f(xi)Δx, where xi is the x-coordinate of the rectangle.
The total area is the sum of the areas of all the rectangles: Σ f(xi)Δx, where the sum is from i = 1 to N.
As Δx approaches 0 (and N approaches infinity), the sum becomes an integral.

Integration

Class 11

⚡ Quick Summary

Integration is like adding up infinitely many tiny pieces to find a total. It's the opposite of differentiation, and we use it to find areas, volumes, and other things by summing up continuous quantities.

∫ab f(x) dx = F(b) - F(a), where dF(x)/dx = f(x)

Integration is the reverse process of differentiation.
If dF(x)/dx = f(x), then F(x) is the integral of f(x).
The integral of f(x) from a to b is denoted as ∫ab f(x) dx.
To evaluate the definite integral ∫ab f(x) dx, find a function F(x) such that dF(x)/dx = f(x), and then calculate F(b) - F(a).