1. (a) Approximate the area under the curve y = x on the interval [1, 2]...

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1. (a) Approximate the area under the curve y = x on the interval [1, 2] using R3, L3, and M3. For each approxi- mation, sketch the corresponding rectangles. 2 1 2 2- 3 (b) Find Rn, Ln, and Mn. What happens to Rn, Ln, and Mwhen n ? 2 2 -1- 3 2 2. Approximate the area under the curve y = x - 1 on the interval [0, 2] by chopping the interval into subintervals of equal width. VVV (a) Find R, L4, M4 and sketch the corresponding rectangles. R L4 M4 (b) Find Rn, Ln, and Mn. (c) Find lim Rn, lim Ln, and lim Mn. n n n Finding Areas using Antiderivatives 1. For each of the function f, find the area A(x) between the graph of f and the interval [-1, x], and find the derivative A'(x) of this area function. (a) f(x) = x + 1. (b) f(x) = 2x + 3. 2. Use the antiderivative method to find the area under the graph of y = x - 1 on the interval [0,2]. 1. (a) Approximate the area under the curve y = x on the interval [1, 2] using R3, L3, and M3. For each approxi- mation, sketch the corresponding rectangles. 2 1 2 2- 3 (b) Find Rn, Ln, and Mn. What happens to Rn, Ln, and Mwhen n ? 2 2 -1- 3 2 2. Approximate the area under the curve y = x - 1 on the interval [0, 2] by chopping the interval into subintervals of equal width. VVV (a) Find R, L4, M4 and sketch the corresponding rectangles. R L4 M4 (b) Find Rn, Ln, and Mn. (c) Find lim Rn, lim Ln, and lim Mn. n n n Finding Areas using Antiderivatives 1. For each of the function f, find the area A(x) between the graph of f and the interval [-1, x], and find the derivative A'(x) of this area function. (a) f(x) = x + 1. (b) f(x) = 2x + 3. 2. Use the antiderivative method to find the area under the graph of y = x - 1 on the interval [0,2].