4.4B THE FUNDAMENTAL THEOREM OF CALCULUS The Second Fundamental Theorem!
Another use for definite integrals Now that we are a little more comfortable with definite integrals and how to evaluate them, we will see another use. Instead of both the upper and the lower limits being constants, we will look at what happens when the upper limit is allowed to vary. Definite integral as a number: f is a function of x F is a function of x f is a function of t Definite integral as a function of x: constant constant constant
Time to explore With your graphing calculator, graph Hint: Y 1 =fnInt(cos x, x, 0,x) and choose window x: [0, π , π /2] & y:[-3, 3,1] Do you recognize this graph? Explain. This is a HUGE idea. It can be really useful, because we can graph F(x) even if we can’t do the integration of what is in the integrand! We can also use it when we want to allow the upper limit to vary, as in the next example.
Ex 6, p. 288 Definite integral as a function Evaluate the function for x =0, π /6, π /4, π /3, and π /2 You can think of F(x) as accumulating the area under curve f(t) = cos t from t = 0 to t = x. This interpretation of an integral as an accumulation function is useful in applications of calculus
We are about ready for the second fundamental theorem of calculus! Notice results from the last example: so Notice that this is the original integrand with “t” replaced with “x”
The second fundamental tells you if a function is continuous, it has an antiderivative . . . . . . but notice, they don’t tell you how to find it, just that it exists, and is graphable! Ex 7 p. 290 Notice that sin t 3 is continuous everywhere So this is very straightforward. Notice that you are finding the derivative of an integral!
Ex 8 p 290 Using the Second Fundamental Theorem of Calculus Letting u = x 3 we will apply the second fundamental theorem along with the chain rule du/dx=3x 2 . . . and Definition of dF/du Substitution for F(x) Substitute u = x 3 Rewrite as function of x
Say what? and found F’(x) We can look at this particular problem from another viewpoint Because we CAN find this integral. Now find F’(x) We just verified that the 2 nd Fundamental theorem is true!
These are just weird enough that you might actually need to study them after you think about them some more! 4.4b p. 291/ 62-64, 67-71 odd, 75-91 odd, 90