Left endpoint approximation

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Appendix Area under a curve, Summary of method using Riemann sums. To find the area under the curve y = f(x) on the interval [a, b], where f(x) ≥ 0 for all x in [a, b] and continuous on the interval: 1. Divide the interval into n strips of equal width ∆x = b−a. This divides the interval [a, b] into n n subintervals: [x0 = a, x1], [x1, x2], [x2, x3], . . . , [xn−1, xn = b]. (Note x1 = a+∆x, x2 = a+2∆x, x3 = a+3∆x, . . . xn−1 = a+(n−1)∆x, xn = a+n∆x = b.) 2. For each interval, pick a sample point, x ∗ i in the interval [xi−1, xi]. 3. Construct an approximating rectangle above the subinterval [xi−1, xi] with height f(x ∗ i ). The area of this rectangle is f(x ∗ i )∆x. 4. The total area of the approximating rectangles is (The sum is called a Riemann Sum.) f(x ∗ 1)∆x + f(x ∗ 2)∆x + · · · + f(x ∗ n)∆x 5. We define the area under the curve y = f(x) on the interval [a, b] as A = lim n→∞ [f(x ∗ 1)∆x + f(x ∗ 2)∆x + · · · + f(x ∗ n)∆x] = lim n→∞ Rn = lim n→∞ [f(x1)∆x + f(x2)∆x + · · · + f(xn)∆x] = lim n→∞ Ln = lim n→∞ [f(x0)∆x + f(x1)∆x + · · · + f(xn−1)∆x]. (In a more advanced course, you would prove that all of these limits give the same number A which we use as a measure/definition of the area under the curve) 6
Extra Example Estimate the area under the graph of f(x) = 1/x from x = 1 to x = 4 using six approximating rectangles and ∆x = b−a n = , where [a, b] = [1, 4] and n = 6. Mark the points x0, x1, x2, . . . , x6 which divide the interval [1, 4] into six subintervals of equal length on the following axis: Fill in the following tables: xi x0 = x1 = x2 = x3 = x4 = x5 = x6 = f(xi) = 1/xi (a) Find the corresponding right endpoint approximation to the area under the curve y = 1/x on the interval [1, 4]. R6 = (b) Find the corresponding left endpoint approximation to the area under the curve y = 1/x on the interval [1, 4]. L6 = (c) Fill in the values of f(x) at the midpoints of the subintervals below: midpoint = x m i x m 1 = x m 2 = x m 3 = x m 4 = x m 5 = x m 6 = f(x m i ) = 1/x m i Find the corresponding midpoint approximation to the area under the curve y = 1/x on the interval [1, 4]. M6 = 7
Page 1 and 2: 2 1.8 Approximating Area under a cu
Page 3 and 4: x ∗ i in each interval to produce
Page 5: 5. Finish the calculation above and
Page 9 and 10: Extra Example Find the area under t

Left endpoint approximation

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