What is an integral?
An integral is the total area under the curve of a function. It is represented by a curve ∫ with an upper and lower limit. These limits determine the x-values at which you are actually taking the area under the curve. Integrals are also considered as “anti-derivates” which mean that you are changing your unit in the reverse direction as you would as if you took the derivative. For example the derivative of the distance traveled by a car is the velocity of the car; this means that the integral of the velocity of the car will give us the total distance travelled by the car. Now that we have a working definition of what an integral is let’s answer some specific questions that people may have when they first start working with integrals.
How does an integral find the area of a function?
When we try to find the area under a curve we would basically take a geometric approach by making a bunch of rectangles, or trapezoids and then adding all of the rectangles area together to get one total. One should note that when creating these rectangles there is error between the edges of our rectangles and the curve of our function that would make us over or underestimate. When creating these rectangles one should also notice that the thinner we make these rectangles the more accurate our results will be. We call this integral approximation and you can see a very thorough visual representation here. An integral has the same method but it creates its rectangles infinitely thin to calculate a very accurate result.
What about the integral of the area under the x-axis?
When taking the integral of the function we are finding the area between it and the x-axis. This means that when we are above the x-axis we are finding the area under the curve; and when we are under the x-axis we are finding the area above the curve. When we are comparing these results we find the area under the curve is positive (+) and the area under is negative (-); this means that when we add these areas together to get our total area. This would be better represented as all of the area above the x-axis minus the area below the x-axis.
Can we find the area between two functions?
Yes, to do this we would take the integral of the upper function and subtract the integral of the bottom function. This means that we would have the area that f1 overlaps – f2 overlaps giving us the area between or below f1 and above f2. You can see a visual modelhere.
Do our integrals always have to be bound?
Yes, if the area does not have a specific boundary or intersection that we can stop calculating the area our function would expand all the way toward infinity. This would mean that since we are covering an infinite amount of x the height of the function would not matter because an infinite base would mean an infinite amount of area.
Now that we have some basic guidelines to follow when thinking about taking the integral of a function we can apply them when looking at specific integral laws. One specific law is the opposite of the power rule: dy/dx(x^n) = n*x^(n-1)*dx. This makes our opposite rule ∫(x^n*dx) = (x^(n+1))/(n+1).Now is not the time to memorize every law but just remember that an integral is an anti-derivative so just work the derivative laws backwards and you will only have a few new laws along with what you already know.