29. Definite Integral

Dated: 30-10-2024

Previously, we sliced the area into rectangles with equal widths. Was it mandatory? no.
Now we will slice them into rectangles with width that are not equal in length.

These slices or subintervals are also called partitians of the interval¹ and the biggest one is called the mesh size, denoted as \(max(\Delta x_i)\).

Here is the important part, if \(max(\Delta x_i) \to 0\) then every \(\Delta x \to 0\).

\[A = \lim_{max(\Delta x_i) \to 0} \sum_{i = 1}^{n}f(x_i) \cdot \Delta x_i\]

Definite Integral of Continuous Function² with Non-negative Values

\[\int_a^b f(x)dx = \lim_{max(\Delta x_i) \to 0} \sum_{i = 1}^{n}f(x_i) \cdot \Delta x_i\]

This left side is called definite integral with \(a\) being its lower limit and \(b\) being its upper limit.

This term \(\sum_{i = 1}^{n}f(x_i) \cdot \Delta x_i\) is called Riemann Sum.

Example

This area can be found out using:

\[\int_1^2 x \cdot dx\]

Definite Integral of Continuous Function²

The net signed area of a continuous function² defined over the interval¹ \((a, b)\) is also found out by

\[\int_a^b f(x) \cdot dx\]

This is the \(\sin(x)\) function.³
Let us say that the area of the blue section is \(c\) then the area of the green section will be \(-c\) (due to symmetric reasons).
Hence, the net area is \(0\)!

\[\int_0^{2 \pi} \sin(x) \cdot dx = 0\]

Properties of Definite Integrals

• \[\int_a^b f(x) \cdot dx = - \int_b^a f(x) \cdot dx\]
• \[\int_a^b c f(x) \cdot dx = c \int_a^b f(x) \cdot dx\]
• \[\int_a^b (f(x) + g(x)) \cdot dx = \int_a^b f(x) \cdot dx + \int_a^b g(x) \cdot dx\]
• \[\int_a^b (f(x) - g(x)) \cdot dx = \int_a^b f(x) \cdot dx - \int_a^b g(x) \cdot dx\]

• If \(a < c < b\) then \(\(\int_a^b f(x) \cdot dx = \int_a^c f(x) \cdot dx + \int_c^b f(x) \cdot dx\)\)

Theorems

1. If \(f(x)\) is integrable on the interval¹ \([a, b]\) and \(f(x) \ge 0\) for all \(x\) in \([a, b]\) then \(\(\int_a^bf(x) \cdot dx \ge 0\)\)

2. if \(f(x)\) and \(g(x)\) are integrable on the interval¹ \([a, b]\) and \(f(x) \ge g(x)\) for all \(x\) in \([a, b]\) then \(\(\int_a^bf(x) \cdot dx \ge \int_a^b g(x) \cdot dx\)\)

3. If \(f(x)\) is a continous function² bound by \([a, b]\) then \(f(x)\) is integrable on \([a, b]\)

4. If \(f(x)\) is a function³ bound by \([a, b]\) with points of discontinuity³ then \(f(x)\) is integrable on \([a, b]\).
5. If \(f(x)\) not bound by \([a, b]\) then \(f(x)\) is not integrable on \([a, b]\)

References

Read more about notations and symbols.

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1. Read more about intervals. ↩↩↩↩

2. Read more about continuity. ↩↩↩↩

3. Read more about functions. ↩↩↩