# Upper and lower sum

Integral calculus is used to **calculate the area** in an interval between the graph of a function and the x-axis.

### Info

**partition of an interval**method, which forms the origin of integral calculus.

If you want to determine the area now, one partitions the area into vertical rectangles. There are two possibilities:

The first partition of the surface is called **lower sum** and is smaller than the area.

This is the **upper sum** which is larger than the actual area.

$\text{lower sum} \le A \le \text{upper sum}$

### Remember

### Example

$f(x)=x^2$ in the interval $[0; 1]$

You can now calculate the areas of the rectangles (width is $0.25$ and height is $x^2$) and get the following:

#### Lower sum

$s=0.25\cdot (0^2+0.25^2+0.5^2+0.75^2)$ $=\frac{7}{32}$

#### Upper sum

$S=0.25\cdot (0.25^2+0.5^2+0.75^2+1^2)$ $=\frac{15}{32}$

#### Result

$\frac{7}{32} \le A \le \frac{15}{32}$

With a higher number of partitions, the result becomes more and more accurate. So at a number of 256 partitions:

$0.331\le A\le 0.335$