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Section 6.2: Area Suppose that f is a continuous function on the interval [a, b]. What is the area of the region S that lies under the curve y = f (x) from x = a to x = b? To approximate the area of this region, we begin by subdividing the interval [a, b] into n subintervals by choosing partition points x0 , x1 , x2 , . . . , xn such that a = x0 < x1 < x2 < · · · < xn−1 < xn = b. Then the n subintervals are [x0 , x1 ], [x1 , x2 ], . . . , [xn−1 , xn ]. This subdivision is called a partition of [a, b], and denoted by P . The length of the ith subinterval [xi−1 , xi ] is ∆xi = xi − xi−1 . The length of the longest subinterval is called the norm of P and denoted by ||P ||. That is, ||P || = max{∆x1 , ∆x2 , . . . , ∆xn }. Then we choose a representative point x∗i in each subinterval [xi−1 , xi ] and construct an approximating rectangle Ri with base ∆xi and height f (x∗i ). The area of the region S is approximated by the Riemann sum A≈ n X f (x∗i )∆xi . i=1 1 Example: Estimate the area under the graph of f (x) = 16 − x2 on [0, 4] using four approximating rectangles and left endpoints. Is your estimate an underestimate or overestimate? 2 Example: Estimate the area under the graph of f (x) = 4 cos x on [0, π/2] using four approximating rectangles and right endpoints. Is your estimate an underestimate or overestimate? 3 Example: Estimate the area under the graph of f (x) = ex on [−2, 2] using four approximating rectangles and midpoints. 4 Theorem: If f (x) ≥ 0 on [a, b], then the true area under the graph of y = f (x) on [a, b] is A = lim ||P ||→0 n X f (x∗i )∆xi = lim n→∞ i=1 n X i=1 f (x∗i ) b−a n . Example: Find the area under the curve y = x2 + 3x − 2 on [1, 4] using a Riemann sum with equal subintervals and right endpoints. 5 Example: Determine a region whose area is equal to the given limit. r n X 3 3i (a) lim 1+ n→∞ n n i=1 n X π πi (b) lim sin n→∞ 4n 4n i=1 6