How does integration work algebraically?

Question

Based on Why do differentiating and integrating 'work'?

In differentiation, you take the gradient of a point $x$ by differentiating. To this, you take two points on the graph, one which is the point where you want to find the gradient of, and another one (which is explained later).

As you can see, there are two points:

$$(a,f(a)) \text{ and } (a+h,f(a+h))$$

Where $h$ is the distance between the points.

We take the gradient of these two points, and we get:

$$f'(x)=\dfrac{f(a+h)-f(a)}{(a+h)-a}$$

$$=\dfrac{f(a+h)-f(a)}{h}$$

To find the gradient of a point, we take $h$ as $h \rightarrow 0$ which gives us the gradient of the graph at $x$

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So, now about integration.

I understand that integration is using a trapezium rule to take the limit as the width of each strip $\rightarrow 0$ and the number of strips $\rightarrow \infty$

This gives us $$ \lim\limits_{n\to\infty} \sum\limits_{i=1}^n f(x_i)\Delta x_i$$

$n=$ number of strips

$\Delta x_i =$ width of strip $i$

So what is the algebraic reasoning behind this transformation of a function?

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Answer 1

Not sure if this answers your question, but this is how I like to think about integration.

Consider a small change of the area under $f(x)$, this is $dA$. Since $dA$ is small, the red part (change in area) approaches roughly $f(x)dx$ (area of rectangle), so $dA=f(x)dx$, or $$\frac{dA}{dx}=f(x)$$

This means that $A(x)$ is the anti-derivative of $f(x)$.

Answer 2

We can estimate differentiation by taking the two points $f(x)$ and $f(x+h)$ like you said, and then shrinking $h$. What algebraic differentiation does is reduces the $h$ to an extremely small quantity, and outputs the gradient between two points that are virtually the same but only just not.

With the Trapezium Rule for Integration Approximation, you have a class width ($h$ or $w$). As your h shrinks, the values get closer. Again, all algebraic integration does is shrink your h to an extremely small quantity and gives you the area there, which due to the tiny class width, is accurate to the curve.