Clicker Question 1 What is ? A. -2/x 3 + tan(x ) + C B. -1/x – 1/(x + 1) + C C. -1/x + tan(x ) + C D. -1/x + arctan(x ) + C E. -2/x 3 + arctan(x ) + C

Clicker Question 2 Estimate using one trapezoid, to three decimal places. A B C D E

The Fundamental Theorem of Calculus (4/13/12) It turns out that we can get the exact definite integral of a function f (x ) on and interval [a, b] provided we can compute an antiderivative of f (x ). One needs only find F (x ), evaluate it at the right-hand endpoint b, evaluate it at the left-hand endpoint a, and subtract the two values!

Statement of the FTC If F (x ) is any antiderivative of f (x ), then So, to get exact answers to integral questions, we need only find an antiderivative and evaluate it twice! This is VERY POWERFUL theorem.

Examples Previously we estimated the area under f (x ) = x 3 on [0, 2] (using 2 trapezoids) to be 5 sq. units. According to the FTC, what’s the exact answer? Does it make sense that it’s smaller? Previously we estimated the area under f (x ) = cos(x ) on [0,  /2] (using 2 trapezoids) to be.95 sq. units. According to the FTC, what’s the exact answer? Does it make sense that it’s larger?

Clicker Question 3 According to the FTC, what is the exact area under f (x ) = e x on the interval [0, 3]? A. e 3 B. 3e 2 C. (1/4)e 4 – 1 D. e 3 – 1 E. (1/4)e 4 – 1

Remarks on the FTC The FTC is not a “magic wand” which will always work. It depends on being able to write down an antiderivative of the given function. This may or may not be possible. What is ?

Remarks (continued) The FTC actually has two parts. This part (called Part 2 by our text) says that you can use the antiderivative (a function) to get the definite integral (a number). Part 1 says if you can turn the definite integral into a function by “freeing up” the right hand endpoint, that function is an antiderivative.

Assignment for Monday On page 395 of our text, do Exercises odd. Test #2 corrections are due Tuesday (4/17) at 4:45.