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1314L19 - f from x = 1 to x = 3 Example 2 Evaluate dx e x x...

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Lesson 19 – The Fundamental Theorem of Calculus 1 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. Example 1: Suppose 1 2 ) ( = x x f . Find the area under the graph of
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Unformatted text preview: f from x = 1 to x = 3. Example 2: Evaluate dx e x x ) 4 6 ( 2 2-∫ Lesson 19 – The Fundamental Theorem of Calculus 2 Example 3: Evaluate: ∫-+ 3 1 2 ) 7 4 3 ( dx x x Example 4: Evaluate: ∫ -4 1 2 1 2 dx x x Lesson 19 – The Fundamental Theorem of Calculus 3 Example 5: Evaluate: ∫ +-5 2 2 6 4 2 dx x x x Example 6: Evaluate: dx x e x ∫ +-3 ) 1 ( Lesson 19 – The Fundamental Theorem of Calculus 4 Example 7: In a recent pretrial run for the world water speed record, the velocity of the Sea Falcon II t sec after firing the booster rocket was given by 440 20 ) ( 2 + +-= t t t v , ) 20 ( ≤ ≤ t ft/sec. Find the distance covered by the boat over the 20-sec period after the booster rocket was activated. 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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