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Unformatted text preview: c Kendra Kilmer April 2, 2009 Section 6.4 - The Definite Integral
Example 1: Find the area under the curve y = x2 + 1 on the interval [0, 2]. 7 c Kendra Kilmer April 2, 2009 Theorem: If f (x) > 0 and is either increasing on [a, b] or decreasing on [a, b] then its LHS and RHS approach the same real number as n . Left-hand sums and right-hand sums are special cases of Riemann Sums where each sum is found using any x-value in the subintervals to determine the heights of the rectangle. Definition: For a continuous function f the definite integral of f on the interval [a, b] is where a and b are the lower and upper limits of integration, respectively. Note: Geometrically, the definite integral represents the cumulative sum of the signed areas between the graph of f (x) and the x-axis from x = a to x = b, where areas above the x-axis are counted positively and areas below the x-axis are counted negatively. Example 2: Calculate the following given f (x) below b a)
0 f (x) dx c b)
0 f (x) dx d c)
a f (x) dx 8 c Kendra Kilmer April 2, 2009 Properties of Definite Integrals
a b f (x) dx = f (x) dx =
a b 2. 3.
a b k f (x) dx = [ f (x) g(x)] dx =
a b 4. 5.
a f (x) dx =
4 4 Example 4: Given
1 4 x dx = 7.5,
1 x2 dx = 21, and
4 5 x2 dx = 61/3, calculate the following: a)
1 (4x2 - 9x) dx 5 b)
1 (-4x2 ) dx Section 6.4 Homework Problems: 1,3,5,17,21,25,29,33,37 and supplemental problems 9 ...
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- Fall '08
- dx, Kendra Kilmer