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Math 142 Lecture Notes for Section 6.6 Section 6.6 - 1 Area Between Two Curves Theorem 6.6.1: If f (x) and g(x) are two continuous functions with f (x) ≥ g(x) on [a, b] , then the area between the two curves on [a, b] is given by Example 6.6.2: Find the area between y = x2 − 3 and the x-axis on [0, 3]. Example 6.6.3: Find the area between y = x and y = x3 + 4 on [0, 4]. Math 142 Lecture Notes for Section 6.6 2 Example 6.6.4: Find the area bounded between the curves y = x3 − 1 and y = x on [−3, 0] Example 6.6.5: Find the area bounded between the curves y = 2x and y = x2 on the interval [0, 4]. If a < b < c < d and f (x) ≥ g(x) on [a, b] and [c, d], with g(x) ≥ f (x) on [b, c] how do we compute the area between f (x) and g(x) on [a, d]? Math 142 Lecture Notes for Section 6.6 3 Example 6.6.6: A colony of bees is growing at the rate of b0 (t) = 300et/5 where t is time in days. Find the area between the graph of b0 (t) and the t-axis on the interval [0, ln 32] and interpret its meaning. Example 6.6.7: √ If two plants are growing at rates of 1 + t and t2 + 1 respectively with t being time in weeks, what are their heights after 1/4 of a week? After 9/4 of a week? Section 6.6 Suggested Homework: 1-35(odd), 41, 51, 59, 61