Lesson 15 - Area by Integration

Lesson 15 - Area by Integration - TOPIC APPLICATIONS AREA...

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TOPIC APPLICATIONS AREA BY INTEGRATION
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The area under a curve Let us first consider the irregular shape shown opposite. How can we find the area A of this shape?
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The area under a curve We can find an approximation by placing a grid of squares over it. By counting squares, A > 33 and A < 60 i.e. 33 < A < 60
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The area under a curve By taking a finer ‘mesh’ of squares we could obtain a better approximation for A . We now study another way of approximating to A , using rectangles, in which A can be found by a limit process.
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The area under a curve The diagram shows part of the curve y = f ( x ) from x = a to x = b . We will find an expression for the area A bounded by the curve, the x -axis, and the lines x = a and x = b . A
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The area under a curve The interval [ a , b ] is divided into n sections of equal width, Δ x . n rectangles are then drawn to approximate the area A under the curve. Δ x A
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The area under a curve Dashed lines represent the height of each rectangle. Thus the area of the first rectangle = f ( x 1 ) . Δ x 1 f ( x 1 ) The first rectangle has height f ( x 1 ) and breadth Δ x 1. The position of each line is given by an x -coordinate, x n . x 1 , x 2 , x 3 , x 4 , x 5 , x 6 Δ x 1
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The area under a curve An approximation for the area under the curve, between x = a to x = b , can be found by summing the areas of the rectangles. A = f ( x 1 ) . Δ x 1 + f ( x 2 ) . Δ x 2 + f ( x 3 ) . Δ x 3 + f ( x 4 ) . Δ x 4 + f ( x 5 ) . Δ x 5 + f ( x 6 ) . Δ x 6
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The area under a curve Using the Greek letter Σ (sigma) to denote ‘the sum of’, we have = = 6 1 ). ( i i i i x x f A = = n i i i i x x f A 1 ). ( For any number n rectangles, we then have
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= = n i i i x x f A 1 ). ( = = a x b x x x f A ). ( The area under a curve In order to emphasise that the sum extends over the interval [ a , b ], we often write the sum as
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= = = a x b x x x f A x ). ( lim 0 The area under a curve By increasing the number n rectangles, we decrease their breadth Δ x . As Δ
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This note was uploaded on 10/19/2011 for the course MATH 22 taught by Professor Ma'amzapanta during the Fall '11 term at Mapúa Institute of Technology.

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Lesson 15 - Area by Integration - TOPIC APPLICATIONS AREA...

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