PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Definite integrals can be used to find the net change in a function between two times. The fundamental theorem of Calculus can be restated so that the definite integral of the function's derivative is equal to the net change in the function between two values. Using definite integrals as net change is an accurate way to compute the net change of a quantity.
I want to talk about how the definite integral can be used to measure a net change of a function. First let's revisit the fundamental theorem of calculus. It says if little f is a continuous function and if big f is an anti-derivative of little f, then the definite integral from a to b of little f of x is capital F of b minus capital F of a. So this is my anti-derivative and I'm evaluating it at b and at a. This is generally how we evaluate definite integrals whenever we can. But there's another way to look at this. First this phrase f is an anti-derivative, capital F is an anti-derivative of little f means capital F prime equals little f of x. So you can rewrite the fundamental theorem in the following way. Just replace the little f of x with capital F prime and it's still true. The integral from a to b of capital F prime of x dx equals capital F of b minus capital F of a. So I want to analyze what this means, what this new equation, this new theorem means. So what does it mean?
Well, first of all, the right hand side, f of b minus f of a is the net change in the value of f of x from a to b. So if f is some amount like say the amount of money in your bank, a and b might be 2 different times. Say b is one year from now and a is now. This difference would measure how much the amount in your bank increased. So this is the net change in the amount in your in the amount of your bank account from time a to time b.
Now, f prime of x is the rate of change of that quantity. The rate of change of f of x. So this, this theorem means the integral of the rate of change of f prime from a to b equals the net change in the function f from a to b. So this gives us a new way of calculating the net change, especially when all we know is about its rate of change. If you don't know what this function is, but you do know what this function is, you can evaluate this difference using this integral. It's a very important idea and we'll look we'll call this the net change theorem. The integral from a to b of capital F prime of x dx equals capital F of b minus capital F of a.