In this topic, we are going to connect the two big ideas in Calculus: instantaneous rate and area under a curve. We'll see that a definite integral can be thought of as an infinite sum of infinitely small things and how this connects to the derivative of a function!
You are very familiar with taking the derivative of a function. Now we are going to go the other way around--if I give you a derivative of a function, can you come up with a possible original function. The symbol which we'll use to denote the anti-derivative will see strange at first, but it will all come together in a few tutorials when we see the connection between areas under curves, integrals and anti-derivatives.
Differential calculus was all about rates (that is, after all, what a derivative is). As we'll see, integral calculus is all about the idea of summing or "integrating" an infinitely many infinitely small small things to get a finite value (often the area under a curve). Despite not really having any calculus in it, this tutorial foreshadows the connection between rates and areas under curves. As we'll see, this is the foundation of the fundamental connections in all of calculus!
In this tutorial, we'll think about how we can find the area under a curve. We'll first approximate this with rectangles (and trapezoids)--generally called Riemann sums. We'll then think about find the exact area by having the number of rectangles approach infinity (they'll have infinitesimal widths) which we'll use the definite integral to denote.
You now know that the area under a curve can be denoted by a definite integral. In this tutorial, we'll extend that knowledge by exploring various properties of the definite integral (that will be really useful later on in life)!
You get the general idea that taking a definite integral of a function is related to evaluating the antiderivative, but where did this connection come from. This tutorial focuses on the fundamental theorem of calculus which ties the ideas of integration and differentiation together. We'll explain what it is, give a proof and then show examples of taking derivatives of integrals where the Fundamental Theorem is directly applicable.
Until now, we have seen definite integrals as the area under a curve. We've approximated this area with reactangles using Riemann sums. We also realized that we could potentially find the exact area if we take the limit as we approach having an infinite, infinitely thin rectangles. But is there an easier way to evaluate an integral? Even more, does this somehow connect to everything we know about the derivative and differential calculus? Hold on to your seats, because everything is about to come together!
Not everything (or everyone) should or could be proper all the time. Same is true for definite integrals. In this tutorial, we'll look at improper integrals--ones where one or both bounds are at infinity! Mind blowing!