For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,6] with 6 subintervals.
A
9.62
B
6.62
C
7.16
D
8.15
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1
Step 1: Understand the problem. We are tasked with approximating the area under the curve f(x) = 3 - √x over the interval [0,6] using a Riemann sum with 6 subintervals and left endpoints.
Step 2: Divide the interval [0,6] into 6 equal subintervals. The width of each subinterval (Δx) is calculated as Δx = (6 - 0)/6 = 1.
Step 3: Identify the left endpoints of each subinterval. The left endpoints are x = 0, 1, 2, 3, 4, and 5.
Step 4: Evaluate the function f(x) = 3 - √x at each left endpoint. This gives f(0), f(1), f(2), f(3), f(4), and f(5).
Step 5: Multiply each function value by the width of the subinterval (Δx = 1) and sum them up to approximate the area under the curve. The Riemann sum is Σ [f(x_i) * Δx] for i = 0 to 5.