Chap6_Sec1

# Chap6_Sec1 - 6 APPLICATIONS OF INTEGRATION APPLICATIONS...

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APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6

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In this chapter, we explore some of the applications of the definite integral by using it to compute areas between curves, volumes of solids, and the work done by a varying force. The common theme is the following general method— which is similar to the one used to find areas under curves. APPLICATIONS OF INTEGRATION
We break up a quantity Q into a large number of small parts. Next, we approximate each small part by a quantity of the form and thus approximate Q by a Riemann sum. Then, we take the limit and express Q as an integral. Finally, we evaluate the integral using the Fundamental Theorem of Calculus or the Midpoint Rule. APPLICATIONS OF INTEGRATION ( *) i f x x

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6.1 Areas Between Curves In this section we learn about: Using integrals to find areas of regions that lie between the graphs of two functions. APPLICATIONS OF INTEGRATION
Consider the region S that lies between two curves y = f ( x ) and y = g ( x ) and between the vertical lines x = a and x = b . Here, f and g are continuous functions and f ( x ) ≥ g ( x ) for all x in [ a , b ]. AREAS BETWEEN CURVES

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As we did for areas under curves in Section 5.1, we divide S into n strips of equal width and approximate the i th strip by a rectangle with base ∆ x and height . ( *) ( *) i i f x g x - AREAS BETWEEN CURVES
We could also take all the sample points to be right endpoints—in which case . * i i x x = AREAS BETWEEN CURVES

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The Riemann sum is therefore an approximation to what we intuitively think of as the area of S . This approximation appears to become better and better as n → ∞. [ ] 1 ( *) ( *) n i i i f x g x x = - AREAS BETWEEN CURVES
Thus, we define the area A of the region S as the limiting value of the sum of the areas of these approximating rectangles. The limit here is the definite integral of f - g . [ ] 1 lim ( *) ( *) n i i n i A f x g x x →∞ = = - AREAS BETWEEN CURVES Definition 1

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Thus, we have the following formula for area: The area A of the region bounded by the curves y = f ( x ), y = g ( x ), and the lines x = a, x = b , where f and g are continuous and for all x in [ a , b ], is: ( ) ( ) f x g x ( 29 ( 29 b a A f x g x dx = - AREAS BETWEEN CURVES Definition 2
Notice that, in the special case where g ( x ) = 0, S is the region under the graph of f and our general definition of area reduces to Definition 2 in Section 5.1 AREAS BETWEEN CURVES

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f and g are positive, you can see from the figure why Definition 2 is true: [ ] [ ] [ ] area under ( ) area under ( ) ( ) ( ) ( ) ( ) b
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## This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

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Chap6_Sec1 - 6 APPLICATIONS OF INTEGRATION APPLICATIONS...

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