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# Calculus

## Area under a curve

Give a function $$f(x)$$ where $$f(x)\geq 0$$  over an interval $$a\leq x \leq b$$, we investigate the area of the region that is under the graph of $$f(x)$$ and above the interval $$[a,b]$$ on the x-axis. For example, the purple shaded region is the region above the x-axis from $$[-1,10]$$ and below the graph of a function $$f$$. Such an area is often referred to as the area under a curve.

However, since the area involves different curves with different arcs, it is difficult to calculate the area. However, if we think of the area as a series of rectangles, we can start to approximate the area.

Let's divide the interval $$[a,b]$$ into $$n$$ subintervals of length $$\Delta x$$ (where $$\Delta x$$  must be $$\frac{b-a}{n}$$). We label the endpoints of the subintervals by $$x_0, x_1, \ldots$$ with the start of the interval as $$a=x_0$$ and the end of the interval as $$b=x_n$$.

Notice $$\Delta x$$ becomes the width of each rectangle, and the height of each rectangle is defined by $$f(x)$$ at the beginning of each subinterval. So we can find the area of each rectangle by multiplying the height $$f(x)$$ by the width $$\Delta x$$. There are four subintervals in the diagram and 5 endpoints ($$x_n$$). So there is one less rectangle then the number of endpoints. If we let represent the number of rectangles, we can estimate the above area by adding the area of rectangles.

$Area \,of\,rectangles=\sum\limits_{i=0}^{n-1} f(x_i)\Delta x$

The above process is what we call a Riemann sum.

## Riemann Sum

The Riemann sum is only an appoximation to the actual area under the curve of the function $$f$$. However, we can improve the approximation by increasing the number of subintervals n, which decreases the width $$\Delta x$$ of each rectangle.

Continuing to increase $$n$$ is the concept we know as a limit as $$n\to\infty$$.

We can then approximate the area under the curve $$A_n$$ as

$A_n = \lim_{n\to\infty} \sum\limits_{i=0}^{n-1} f(x_i)\Delta x$

The above limit is also what we call the definite integral of $$f$$ from $$a$$ to $$b$$.