Integral Calculus Physics Notes

Integration: Integration is just opposite to differentiation, if the functions F(x) and f(x) are related as :
\(\frac{d}{d x}\){F(x)} = f(x) then, ∫ f(x)dx = F(x)
∫f(x) dx is called the integral of f(x).
f(x) Δx is the area of the small segment whose height is f(x) asd width is x. When Δ x → 0 then the sum of such sLnall segments become the integral.
Integral Calculus Physics Notes 1
When the limits of the integral are definite (from x = x1 to x = x2), then,

So, ∫^x₁_x₂ f(x)dx is the process of summation in definite limits x₁ of the continuous function f(x) w.r.t the variable.

Indefinite Integration:
We know that the differentiation of every constant is zero.
∴ \(\frac{d}{d x}\){F(x)} = f(x) and, \(\frac{d}{d x}\){F(x) + constant} = f(x)
Integral Calculus Physics Notes 4
Definite Integral (from a to b)

So, the integration of a constant becomes indefinite.
∫f(x)dx = F(x) + Integration constant

Definite Integration:
A definite integral has start and end values. In other words, there is an interval (say a to b). The values of the interval are placed at the bottom and top of the integral as:
∫^b_a f(x)dx (if the lower limit is a and the upper limit

To find out the definite integral, we subtract the integral at points a and b.
∫^b_a f(x)dx = [F(x)]^b_a = [F(b) – F(d)] …(i)

In other words, we can say that the definite integral between a and b is the indefinite integral at b minus the indefinite integral at a.

It is to be noted that the integration constant is not written after calculating the definite integral.

Applications of Integration:
A large number of problems can be solved through integration. Some of them are mentioned below:

The area between curves: We can find out the area between the two functions by integrating the difference between them.
The average value of a function.
Arc Length: We can use integration to find the arc length of a curve. It can be used by up an infinite number of infinitely small line segments.
Volume of solids with known cross-sectional area.
Area defined by polar graphs: We cannot only find out the area in cartesian coordinates but also in polar coordinates.