Fundamental Theorem of Calculus: Makes a connection between Indefinite Integrals (Antiderivatives) and Definite Integrals (“Area”) Historically, indefinite integration has always been defined to be the inverse of differentiation. is the collection of all possible anti-derivatives of f(x), which happen to differ only by a constant. But definite integration, motivated by the problem of finding areas under curves, was originally defined as a limit of Riemann sums. Is the limit of any Riemann sum as the number of rectangles approaches infinity … provided the limit of the Lower and the limit of the Upper Riemann sums are equal. Sec4-4: (Day1) Fundamental Theorem of Calculus Sec4-4: #2-38 evens

Only later was it discovered that the limits of these Riemann sums can actually be computed with antiderivatives, leading to our modern Fundamental Theorem of Calculus. _______________________________ _________________ The fundamental theorem allows us to calculate definite integrals By using anti-derivatives (indefinite integrals) _________________

Examples: Applying the Fundamental Theorem of Calculus Section 4-4

Examples: Applying the Fundamental Theorem of Calculus

4-4 (Day2) Mean Value Theorem and Definite Integrals You can calculate the exact value of a definite integral by using ONE rectangle. To find the ONE sample point, x=c, for the rectangle apply the Mean Value Theorem to g(x), an anti-derivative of f(x). Example: Find the exact value of the definite integral using one rectangle. To find the ONE sample point, apply the MVT to g(x) over the interval [1,2 ] Section 4-4 (Day2): #36-56 evens, 66, evens

Use the sample point, x = c ≈ 1.53 to find the height of the rectangle. To find the ONE sample point, apply the MVT to g(x), the anti-derivative of f(x), over the interval [1,2 ] Graph of g(x) belowGraph of f(x) below

Use the sample point, x = c to find the height of the rectangle. To find the ONE sample point, apply the MVT to g(x), the anti-derivative of f(x), over the interval [0,2 ] Graph of g(x) below Graph of f(x) below Draw the rectangle Draw the parallel lines YOU TRY THIS ONE!

Mean Value Theorem for Integrals If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that: This theorem just says that you can find the exact value of a definite integral by using one rectangle. You have just learned that to find the sample point, c, you have to apply the mean value theorem to g(x), an anti-derivative of f(x). The idea of Average Value of a Function on an Interval follows naturally from this theorem. Definition of the Average Value of a Function on an Interval If f is integrable on the closed interval [a,b], then the average value of f on the interval is It is clear that the average value is the height of the rectangle. Now we can use the Fundamental Theorem, rather than the MVT, to calculate this value.

Examples: Find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value.

Example: (a) Integrate to find F as a function of x and then (b) demonstrate the Second Fundamental Theorem of Calculus by differentiate the result in part (a). Theorem: The Second Fundamental Theorem of Calculus If f is continuous on the open interval containing a, then the derivative of F(x) is: Example: Use the Second Fundamental Theorem of Calculus to find

(1) Write down the definition of the definite integral. (2) Write down the definition of the indefinite integral. Proof of The Fundamental Theorem of Calculus

(2) Write down the definition of the indefinite integral. (1) Write down the definition of the definite integral. Proof of The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus Mean Value Theorem: If (1) g is a continuous on a/an __________________ interval (2) g is differentiable on a/an __________________ interval. Then, there is a mean value, x = c, in the open interval (a,b) such that (3) How can you be sure that the mean value theorem applies to the function

Exploration 5-6b: The Fundamental Theorem of Calculus Mean Value Theorem: If (1) g is a continuous on a/an closed interval (2) g is differentiable on a/an open interval. Then, there is a mean value, x = c, in the open interval (a,b) such that (3) How can you be sure that the mean value theorem applies to the function Because g is an antiderivative of f, then g has a derivative … Recall, differentiablility implies continuity. So if g has a derivative, it IS ALSO continuous.

Exploration 5-6b: The Fundamental Theorem of Calculus ( 4) The figure shows function g in problem 2. Write the conclusion f the mean value theorem as it applies to g on the interval from x = a to x = x 1, and illustrate the conclusion on the graph.

Exploration 5-6b: The Fundamental Theorem of Calculus ( 4) The figure shows function g in problem 2. Write the conclusion f the mean value theorem as it applies to g on the interval from x = a to x = x 1, and illustrate the conclusion on the graph. Slope of tangent line = slope of secant line

The Fundamental Theorem of Calculus ( 5) The figure shows function f (x) from Problem 2. Let c 1, c 2, c 3, …, c n be the sample points determined by the mean value theorem as in problem 4. Write a Riemann sum R n for Use these sample points and equal  x values. Show the Reimann sum on the graph.

Exploration 5-6b: The Fundamental Theorem of Calculus ( 5) Write a Riemann sum R n for

Exploration 5-6b: The Fundamental Theorem of Calculus By the mean value theorem: By the definition of indefinite integrals, Slope of tangent = Slope of Secant g is an anti-derivative of f if g ’(x)= f (x) On a separate sheet of paper, write down what is on this page and fill in the blanks

The Fundamental Theorem of Calculus ( 6) By the mean value theorem: By the definition of indefinite integrals, By appropriate substitutions, show that R n from problem 5 is equal to: Make a substitution into the Riemann sum we wrote in problem 5.

The Fundamental Theorem of Calculus (6) By appropriate substitutions, show that R n from problem 5 is equal to: Make a substitution into the Riemann sum we wrote in problem 5.

The Fundamental Theorem of Calculus After canceling the  x Rearrange the terms so you can see what will cancel Cancel everything that will cancel to get …

The Fundamental Theorem of Calculus (7) R n from Problem 6 is independent of n, the number of increments. Use this fact, and the fact that L n < R n < U n to prove that the fundamental theorem of calculus: Since R n =g(b)-g(a) Since L n < R n < U n

The Fundamental Theorem of Calculus (8) The conclusion in Problem 7 is called the fundamental theorem of calculus. Show that you understand what it says by using it to find the exact value of: Provide that g is an antiderivative of f 1 st Find g(x), an antiderivative of f 2 nd Evaluate g(x) at a=1 and at b=4 3 rd Subtract to get the exact value: Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (8) Show that you understand what it says by using it to find the exact value of: Provide that g is an antiderivative of f Fundamental Theorem of Calculus