![Ex 3 p284 Using the Fundamental Theorem for Area Notice that the function lies above the x-axis for interval [0, 1] so result can be thought of as area. Find the area bounded by graph of f(x) = 2x 2 - 3x + 2, y=0, x=0, and x = 1](https://image.slidesharecdn.com/calc4-4a-111205233858-phpapp02/85/Calc-4-4a-8-320.jpg)
![The value of f(c) from the Mean Value Theorem ends up being called the Average value of f on the interval [a, b].](https://image.slidesharecdn.com/calc4-4a-111205233858-phpapp02/85/Calc-4-4a-10-320.jpg)
Calc 4.4a
- 1. Evaluate a definite integral using the fundamental theorem of calculus Understand and use the mean value theorem Find the Average Value of a function over a closed interval
- 2. You can see that differentiation and integration are inverse relationships to each other. One talks about the quotient ∆y/∆x or a “slope” concept, and one talks about the product ∆y ∆x, or an “area” concept.
- 3. Early mathematicians didn’t know that indefinite integration and definite integration were related. But we get to understand the relationship today. Drum roll, please!
- 4. Guideline #1 is a biggie. Integration requires a rule to apply, and even simple functions sometimes can’t be integrated
- 5. Ex 1 p 284 Evaluating a definite Integral
- 6. Ex 2 p284 A definite integral involving absolute value The vertex is (1/2, 0) so split up integral
- 7. I know it is easy to make little mistakes on this evaluation process. You need to show the integration step , but the evaluation step you can do by calculator unless exact answers are required. Two ways: On main calculator screen: MATH fnInt ( expression, variable, lower limit, upper limit ) On graph screen: Graph function as usual, then select 2 ND CALC, ∫f(x)dx, select lower limit, upper limit, and enter. Both of these can give rounding errors, so know what kind of answers you expect and think accordingly.
- 8. Ex 3 p284 Using the Fundamental Theorem for Area Notice that the function lies above the x-axis for interval [0, 1] so result can be thought of as area. Find the area bounded by graph of f(x) = 2x 2 - 3x + 2, y=0, x=0, and x = 1
- 9. = height x base I imagine the area above f(c) in pink pouring into the blue area to level things out.
- 10. The value of f(c) from the Mean Value Theorem ends up being called the Average value of f on the interval [a, b].
- 11. Ex 4 p 286 Finding the average value of the function over the given interval. Now that we’ve got it, what is it?
- 12. 4.4a p. 291/ 1-45 every other odd, 47, 49, 55-60