6.1: Areas Between Curves (Dated: September 26, 2011) The area A of the region bounded by the curves y = f ( x ), y = g ( x ), and the lines x = a and x = b , where f and g are continuous is given by A = Z b a | f ( x )-g ( x ) | dx . This can be obtained by considering the deﬁnition of the integral on the right-hand side in terms of Riemann sums. Note that A = Z b a [ f ( x )-g ( x )] dx if f ( x ) ≥ g ( x ) for all x in [ a , b ]. Likewise, the area A of the region bounded by the curves x = f ( y ), x = g ( y ), and the lines y = c and y = d , where f and g are continuous is given by A = Z d c | f ( y )-g ( y ) | d y . Example 1 (Selected exercises from exercises 6.1.5-6.1.28) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then ﬁnd the area of the region. 6: y = sin x, y = e x , x =0 , x = π 2 Ans: e π /2-2 8*: y = x 2-2 x, y = x + 4 Hint: The curves intersect when
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