Lesson 19-filled - Example 1 : Suppose 1 2 ) ( + = x x f ....

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Math 1314 Lesson 19 The Fundamental Theorem of Calculus In the last lesson, we approximated the area under a curve by drawing rectangles, computing the area of each rectangle and then adding up their areas. We saw that the actual area was found as we let the number of rectangles get arbitrarily large. Computing area in this manner is very tedious, so we need another way to find the area. The fundamental theorem of calculus allows us to do just this. It establishes a relationship between the antiderivative of a function and its definite integral. The Fundamental Theorem of Calculus Let f be a continuous function on [ a, b ]. Then ) ( ) ( ) ( a F b F dx x f b a - = where F ( x ) is any antiderivative of f . If you are interested in seeing why this works, see the link for the “proof” of the fundamental theorem of calculus on Marjorie Marks’ class notes page.
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Unformatted text preview: Example 1 : Suppose 1 2 ) ( + = x x f . Find the area under the graph of f from x = 1 to x = 3. Example 2 : Evaluate dx e x x ) 4 6 ( 2 2- Example 3 : Evaluate: -+ 3 1 2 ) 7 4 3 ( dx x x Example 4 : Evaluate: 4 2 1 1 1 dx x x - Example 5 : Evaluate: +-5 2 2 6 4 2 dx x x x Example 6 : Evaluate: dx x e x +-3 ) 1 ( Example 7 : Evaluate: ( 29 ( 29 dx x x 1 3 2 2-- Example 8 : Find the area of the region under the graph of 2 3 ) ( x x x f-= over the interval [0, 3]. From this lesson, you should be able to State the fundamental theorem of calculus Use the FTOC to compute definite integrals Use the FTOC to find area under a curve...
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This note was uploaded on 02/21/2012 for the course MATH 1314 taught by Professor Marks during the Fall '08 term at University of Houston.

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Lesson 19-filled - Example 1 : Suppose 1 2 ) ( + = x x f ....

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