Integration in Calculus: Understand What It Is

[ubergewehr273]

I have seen in a lot of textbooks this funny curly bar which denotes integration with a lot of fancy numbers around.
Could anyone tell me what exactly is integration in calculus?

 

[gopher_p]

https://www.google.com/webhp?source...S440US440&ion=1&espv=2&ie=UTF-8#q=integration

https://en.wikipedia.org/wiki/Integral

 

[AMenendez]

An integral is a very, very close approximation to a sum. Read gopher_p's link above, and you'll see that an integral is really a way of approximating the area under a complex curve by summing the areas of appropriately drawn rectangles under the curve. As we make the rectangles thinner and thinner (i.e., $\Delta x \to 0$), then the approximation is more and more accurate because we can draw more rectangles and position them more appropriately. The process I'm talking about is called a "Riemann Sum", which is where the integral comes from. The "curly bar" you're referring to is actually made to look like an elongated "S", as if to stand for "sum".

More succinctly,

$S = \lim_{\Delta x_i \to 0} \sum_{i = 1}^{n} f(x_i) \Delta x_i$, which is called the Riemann integral over an interval $[a, b]$ if the limit exists.

Basically, what it says is if we let $\Delta x_i$, the "base" of the rectangle, get smaller and closer to zero while letting the "height" of the rectangle, $f(x_i)$, the function value, stay the same, then our sum gets more and more accurate.

 

[jack476]

The integral F(x) of a function f(x) is the function for which f(x) is the derivative. The derivative of f(x), which might be called f'(x) or df/dx, is a function that tells you how the value of f(x) changes at any value of x. If you look at the graph of f(x) = x, you'll see that the function changes its value at a constant rate, 1 in this case, and so the derivative of x is 1. If you're still in early calculus, you can think of the derivative as the function that gives you the slope of f(x) at any value of x.

Basically you integrate f(x) and get F(x), and if you take the derivative of F(x) you get f(x).

It can also be thought of as the area under the graph of a function. For instance, if you have f(x) = x, you take the integral and get (1/2)x^2. Note that the graph of f(x) = x forms a triangle. Well, the area of a triangle is 1/2* base* height, and the triangle formed by the graph of f(x) = x forms a triangle where base = height = x, substituting those values, you get (1/2)x^2.

I don't really like that explanation since it doesn't truly show how powerful integration is, but from a purely practical perspective that's a good way to think about it when you first encounter them.

 

[1MileCrash]

AMenendez: the integral is not a very close approximation, it is exact.

The integral of sin(x) from 0 to 42 doesn't give me a very close approximation, it gives me the exact area under the curve.

A Riemann sum gives an approximation to the area of the curve. The limit of the Riemann sum is the integral, and it is not very close, it is exact.

 

[AMenendez]

AMenendez: the integral is not a very close approximation, it is exact.

The integral of sin(x) from 0 to 42 doesn't give me a very close approximation, it gives me the exact area under the curve.

A Riemann sum gives an approximation to the area of the curve. The limit of the Riemann sum is the integral, and it is not very close, it is exact.

Sorry, that's what I meant to say actually. The integral is basically like the continuous counterpart of the discrete Riemann sum, so it is exact--my reasoning was backwards.