Integration – Adding Up the Values of a Function (4/15/09) Whereas the derivative deals with instantaneous rate of change of a function, the (definite)
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Integration – Adding Up the Values of a Function (4/15/09) Whereas the derivative deals with instantaneous rate of change of a function, the (definite) integral deals with adding up all the values of a function over a specific interval. “Integrate” (in mathematics) means “add up.”
Example: Area Find the (exact) area under f (x) = x + 4 on the interval from 0 to 2. No need for calculus here. Find the (exact) area under f (x) = x 2 + 1 on the interval from 0 to 2. (??????) Well, can we try to estimate the area by adding up small approximating pieces? How can we get better and better estimates? (Machines can help.)
Using Antiderivatives To Get Exact Answers! What is an antiderivative F of f (x) = x 2 + 1 ? Note that f is the rate of change of F, so if I add up all of these rates of change over the interval [0, 2], I should get the total change in F. But this is total change is F (2) – F (0). Calculate it.
Clicker Question 3 If F is an antiderivative of f (x ) = x 3, then the area under x 3 from 0 to 2 will be exactly F (2) – F (0). What is this? A. 0 B. 4 C. 8 D. 16/3 E. 16
Clicker Question 4 If F is an antiderivative of f (x ) = sin(x ), then the area under sin from 0 to will be exactly F ( ) – F (0). What is this? A. B. -1 C. -2 D. 2 E. / 2
Definite Integrals How can we “add up” the values of a function over some fixed interval? This is called the definite integral of the function on that interval. One approach is to estimate the sum by taking a finite number of values of the function and giving each value an appropriate “weight”. This is called using Riemann sums to estimate the definite integral. The more values you use, the closer you get the actual answer.
Common Riemann Sums There are three very common Riemann sums which we shall use: The left-hand sum. That is, use the values of the function which lie at the left-hand edges of short subintervals. The right-hand sum. Same idea. The average of these two. This is called the trapezoid rule, and is quite accurate for most functions even using a small number of subintervals.
Notation and Assignment The most common notation for the definite integral of f (x) on the interval [a, b] is On Friday, we’ll start in the classroom but then have Lab #7. In preparation please read Section 5.1 through page 361 and do Exercises 1a and 4. Monday’s assignment will be handed out at lab on Friday.