Understanding Integration, Integrals, Antiderivatives, and their Relationship to Derivatives: the Fundamental Theorem of Calculus

So…what are integrals and how are the related to derivatives? Think of dx, the symbol for infinitesimal change in the x coordinate (it’s different from just delta x in that in dx is the limit as delta x approaches zero) , here in the context of derivatives:

The idea…is to compute the rate of change as the limiting value of the ratio of the differences Δy / Δx as Δx becomes infinitely small.In Leibniz’s notation, such an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written dy/dx. (Wikipedia)

Apparently many definitions of what exactly dy/dx means are lacking (there’s a whole article on JSTOR entitled What Exactly is dy/dx?)  There are manipulations you will need to know with dy/dx.  So, say you have y=f(x), and the derivative with respect to x is  dy/dx=f'(x).  You could then write dy=f'(x)dx.  Then if you take the integral of both sides of this, you get y=f(x)!  So if you had y=ln(x), and wanted to find the integral of this (the antiderivative of natural log), you would write integral of y=integral of ln(x)dx, and you could use integration by parts to solve this–setting u = ln(x), then du = (1/x) dx, dv = dx, and v = x.

With derivatives, the derivative is basically a slope, a rate of change, the change in the y values of a function as the x value, the input of the function, is changed. The derivative is basically an instantaneous rate: we look at a change in x value so small, infinitesimally small, so that you get the slope, the rate of change, of a function at a single x input. The instantaneous rate of change for an x input, the derivative, is a tangent line to the function at that point. Think about the rate of change in an intuitive way: if between two different x values there is no change in the y values, the rate of change is 0, no change in y based on change in x. If there is a positive change in the y values based on change in x inputs, there’s a positive derivative; if there’s a negative change, there’s a negative derivative. For functions that are not linear, the derivative can be zero, positive, and negative across different x values as the curve changes. See this post on critical points, points of inflection, and concavity for more information.

Infinitesimal change in the x input, dx, plays an important role in integrals as well. Whereas with derivatives you are finding the rate of change of a function at a single point, the instantaneous rate, with integrals you are finding the area under a curve between two points. The notation is like \int_a^b f(x)dx. Here, we’re finding the area under the curve between x=a and x=b, where the curve is made by the function f(x). See this article on Riemann Sums for more information and this section on “Areas and Integrals” in Mathematics for Economists (Google Book Search free digitized version).

It’s easy to think about integrals and the area under the curve when you think of the area of a rectangle on a graph. Think about the area under a curve from x=0 to x=5 where the y value is 4 at each x value; think of the area of the space sketched out by the ordered pairs (0,4) and (5, 4). Well, you know the area of a rectangle is computed by base * height. So, it’s easy to see here that the area under the curve is simply x * y=5*4=20. Well, in this case the height is uniformly 4, so that’s easy. But what about when the height varies per x, as in a curve? That’s where Riemann Sums and dx come in.

Basically, think of using tons of very skinny rectangles to approximate the area under a curve; the very skinny rectangles would do a good job of approximating the area under the curve when you added them all together. Well, that’s what happens with integrals. You’re multiplying the y value (the height=) at each new x by the change between each of the x’s, where the change is the infinitisemally small dx. So you have very small changes in x, dx’s, multiplied by the y value at each of those x’s, and you add them all up, and that’s the area under a curve.

Now, how does this relate integrals to derivatives? There’s the Fundamental Theorem of Calculus. Basically, it casts integrals in terms of antiderivatives. Say you have a function f(x), the integral of this function, F(x), is called the antiderivative. When you take the derivative of an antiderivative, you end up with the original function, f(x). That’s the (first?) Fundamental Theorem of Calculus; taking the derivative of an antiderivative reverses the antidifferentiation and you end up with the original equation f(x). It may be specific to indefinite integrals in this part, I’ll look into it more:

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration[1] can be reversed by a differentiation.

The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals. (Wikipedia)

Think of \int_a^b f(x)dx. That’s the area under the curve from a to b. Now think of \int_a^c f(x)dx. That’s the area under the curve from a to c. Now, if you want to find the area under the curve from b to c, you can subtract, \int_a^b f(x)dx - \int_a^c f(x)dx. Well, this is a key part of the Fundamental Theorem of Calculus. Say y=f(x). Say the antiderivative of that is z=F(x). The Fundamental Theorem of Calculus says that dz/dx(F(x))=y=f(x). Another way of writing this is lim h->0 (F(x+h) -F(x))/h = f(x), as shown on p. 375 of the first edition of Mathematics for Economists (Google Book Search digitized version).

Now, y at an x input value is the height of the curve at that x value. Think of the Riemann Sums explanation of integration; if you made the rectangle skinny enough in the x dimension, if the difference in the x values is infinitesimally small, dx, then you basically get one height, one y value. This is how integration and derivatives fit together.

Take a look at the graph at the top of the post again.

Basically, if you have \int_a^b f(x)dx and \int_a^c f(x)dx, the difference between the areas will be \int_a^b f(x)dx - \int_a^c f(x)dx. That leaves you with the area under the curve between b and c. Say c-b=h, and make h infinitesimally small, like dx. h is then dx, your infinitesimally small change in x. Think of the definition of derivatives via difference quotients. Here the area between b and c is z=Area=A(x)=F(c)-F(b)=F(b+h)-F(b), where F depends upon y=f(x); for each difference in x, for each little rectangle, the Area is y * difference in x, and to get the Area z, all of these little rectangles are summed up; look again at the section on Riemann Sums and the definition of integrals.

Now, if you took the derivative of this area function, that would be change in the dependent variable, z over change in the indepenedent variable, x, where the change in x is h. So that would be dz/dx= (F(b+h)-F(b))/dx=(F(b+h)-F(b))/h.

As h is infinitesimally small, the Area function with output z here looks at one rectangle for F(b+h)-F(b), with one y value and one dx value which is h. Looking at the definition of an integral, the one y value here is y=f(x), and there is one integral definition for this one rectangle, \int_b^{b+h} f(x)dx,where b+h=c, so we could also write it as \int_b^c f(x)dx.

So here dz/dx is equivalent to dy/dx here, which is (F(b+h)-F(b)/h.

Now, if the change in x, which here is h, is infinitesimally small, you basically get one y value, and the area is y*dx, where the integral is \int_b^{b+h} f(x)dx, the area is basically f(x)dx, here f(b)dx, which equals y*dx. As dx=h, that is y*h. So the deriviative (F(b+h)-F(b)/h would be equivalent to (y*dx)/dx=(y*h)/h=y. So tada, that’s why if F(x) is the antiderivative of f(x), when you take the derivative of F(x) you are left with y, where y=f(x). Yay! This could be slightly off/inaccurate, so check the Wikipedia post Fundamental Theorem of Calculus and pp. 375-377 of the first edition of Mathematics for Economists for more info (unfortunately p. 375 is blocked out on the Google Book Search digitized version).

Calculus for Dummies has some good graphs illustrating some of these principles such as on pp. 242 and 247.

Here’s the text on the illustration above, in case it’s too small for you to read:

As dx is infinitisemally small,f(b)~f(b+h)=f(c). The area under the curve
between b and b+h is F(b+h)-F(b) (where F(b+h) is the area under the curve
from a to c=b+h, and F(b) is the area from a to b. Since h is so small, the
area is basically f(b)*h. Now, the derivative of this integral area function is
the change in output, area, which is f(b)*h, over change in independent
variable, h. So that equals f(b), which shows the Fundamental Theorem of
Calculus!

Once you start doing integrals there are lots of techniques like partial fractions etc that will make it easier to solve integrals.

Internal tag: math

Some latex instructions, including how to make integral signs, here.

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2 Responses

  1. Yr welcome to make a better, more concise explanation or link to a better one, I’m not like a math teacher or anything

  2. This is the only explanation I’ve read that makes sense. Thanks soo much!

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