For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,5] with 5 subintervals.
A
5
B
215
C
6.25
D
10
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1
Step 1: Understand the problem. We are tasked with approximating the area under the curve f(x) = (1/2)x over the interval [0,5] using a Riemann sum with 5 subintervals and left endpoints.
Step 2: Divide the interval [0,5] into 5 equal subintervals. The width of each subinterval (Δx) is calculated as Δx = (5 - 0)/5 = 1.
Step 3: Identify the left endpoints of each subinterval. The left endpoints are x = 0, 1, 2, 3, and 4.
Step 4: Evaluate the function f(x) = (1/2)x at each left endpoint. This gives f(0) = 0, f(1) = 0.5, f(2) = 1, f(3) = 1.5, and f(4) = 2.
Step 5: Compute the Riemann sum by multiplying the function values at the left endpoints by the width of the subintervals (Δx = 1) and summing them: Riemann sum = Δx * [f(0) + f(1) + f(2) + f(3) + f(4)].