Riemann Integral

Overview


The Riemann integral was one of the first and simplest integrals to be defined.

Definition


Properties


Linearity
{% \int_a^b c_1f(x)+ c_2g(x) dx = c_1\int_a^b f(x) dx + c_2\int_a^b g(x) dx %}
Additivity
{% \int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx %}

Example


The example below demonstrates the lower and upper Riemann sums for the following function:
{% y = 4 \times (x - \frac{1}{2}) ^2 %}
Riemann Lower Sum The Riemann lower sum approximates the integral by calculating the area of the rectangles below the curve for the given mesh size.

Number of Divisions
Riemann Upper Sum The Riemann upper sum approximates the integral by calculating the area of the rectangles above the curve for the given mesh size.

Number of Divisions

Contents