Definite Integrals

There is no question that some terms in the field of mathematics can be a bit intimidating. However, once you get past the names, the way the actual math works is pretty straightforward.

Understanding Integration

Integration is a method used to add slices to find the whole. It can be used to determine central points, volumes, and areas of several useful things. However, the easiest place to begin is with the area under the curve of a function.

While you can calculate the function at several points and add the slices of width Δx; however, the answer you arrive at won’t be very accurate. A better method is to make Δx smaller and add up the smaller slices, which will produce an answer that is a bit more accurate. As the slices get closer to zero in width, you get closer to the true answer.

At this point, you may be thinking – “wow, that’s a lot of adding.”

Don’t fret! You don’t have to add them up because there is a shortcut you can use. Finding the integral is the reverse of finding the derivative.

For example, what is the integral of 2x? You already know the derivative of x2 is 2x, so the integral of 2x is x2.

The symbol used for an integral is a “fancy S”, which indicates the sum for adding up your slices. After the integral symbol, insert the function you want to find the integral for and then finish with dx, which means the slices move in the x direction. This is how you write the answer:

ᶴ 2x dx = x2 + C

What’s the C? It is the “constant of integration” and is in the equation because of all the functions that have a derivative of 2x.

The derivative of the equation x2 + 4 is 2x and the derivative of x2 +99 is 2x and it goes on and on because the derivative of any constant is zero. As a result, when the operation is reversed to determine the integral, you only know 2x the constant could have been any value. As a result, the idea is ended by just writing + C to. A simpler way to understand is – just go with it!

With this equation, you are now ready to find the definite integrals of trig functions. All you have to do is plug in the appropriate values.

Derivatives of Trig Functions

In trigonometry, functions are a useful part of day to day life. However, you may wonder how to find the derivatives of a trig function. To do this, you must use limits. Understanding limits can take time, but if you really want to determine the derivatives of a trig function, it is a skill you need to dive into.

If your head is spinning, don’t worry, you aren’t alone. The good news is, the more you practice, the better you will get at working with these numbers and equations. The bad news is, well, it’s math. More good news – calculators!