Your Integral Calculus Toolbox

Updated: Jan 12, 2020

My favorite metaphor for a math and science education is the toolbox metaphor: every technique you learn is like a tool, which can be used to solve a number of problems in the future. Some tools are very niche, and are used only in rare circumstances; others are so frequently applicable that you don’t even think about them anymore.


You can think of multiplication as your trusty hammer, while derivatives are more like a… um… a pickaxe. Don’t think too hard about it.


When I depict derivatives with this specific pickaxe, suddenly they look so much more useful, don’t they? Buckle up though, because knowing how to differentiate stuff will come in handy today.


Like I said, don’t think too hard about it.


Don’t think about hitting your homework with a pickaxe.

Jokes aside, there’s a whole bunch of stuff to learn about integral calculus, and I hope that by the end of this article you’ll have a better understanding of the toolset you’re being asked to use in this class. The tools described below should help you master roughly the first half of MAT 17B (applies to 16B and 21B as well), depending on your professor. If you don't go to UC Davis, this will help you in your Calculus I or II class (depending on when you learn integrals).


Tool 0: What is an Integral?


At some point, someone told you that an integral is “the area under the curve”. And, at least technically, that person is not wrong. But “the area under the curve” is a bad definition to have in your head about an integral. In reality, integration is the act of adding an infinite number of tiny things together—normally getting a finite solution. Pretty wild, right? This definition will help you with the more visual problems in integral calculus, because when you get stuck, you can always ask yourself: “Which tiny things am I adding together in this problem?” Once you can answer that question, you should know what to integrate.


Tool 1: The Basic Antiderivative


I assume you already know this one: an integral is the opposite of a derivative. (Did the name give it away?) That means that if the derivative of (x^2) is (2x), then the integral of (2x) must be (x^2). You should be able to do more versions of this antiderivative in your head quickly:

$$int 99 x^{100} d x=frac{99}{101} x^{101}$$

If you’re having trouble convincing yourself that the above equation is true, then spend some time with more problems like it. There is no shortcut around using this tool—it is the one you are most likely to continue using after this class is over.


Tool 2: Basic Properties of Integrals


There are a few main points to discuss here. First of all, if you see that an integral consists of two terms added together, you can break the terms apart into two separate integrals. In other words,

$$int cos (x)+sin (x) d x=int cos (x) d x+int sin (x) d x$$

Notice that you cannot do the same thing with multiplication.

$$int x^{2} d x neq int x d x * int x d x$$

If you don’t believe me, just evaluate both sides of the equation above according to the basic antiderivative. They’re not equal. However, this does not mean that all hope is lost for multiplication in integral calculus. If your integral is multiplied by some constant (which is not dependent on your variable of integration), then you can pull it out into the front of your integral.

$$int k * sin (x) d x=k * int sin (x) d x$$

The final thing to mention here is about even and odd functions. As you may know, even functions have symmetry about the y-axis, which means that anything happening at (x>0) is also happening at (x <0 ). Meanwhile, odd functions have symmetry about the origin, which means that anything happening at (x > 0) is flipped over the x-axis for (x < 0). From these definitions of even and odd functions, you can likely convince yourself of the following two truths:

(int_{-a}^{a} y(x) d x=2 * int_{0}^{a} y(x) d x) For an even function

(int_{-a}^{a} y(x) d x=int_{0}^{a} y(x) d x-int_{0}^{a} y(x) d x=0) For an odd function