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info@snapandsolve.comLeft-endpoint estimate: Use the left endpoints of each subinterval to approximate the area. The left endpoints are at x = 1, 2, 3, and 4 with corresponding y-values of 5, 3.5, 2.5, and 2.2. The width of each subinterval is 1. Therefore, the left-endpoint estimate is:
A = 1*(5) + 1*(3.5) + 1*(2.5) + 1*(2.2) = 5 + 3.5 + 2.5 + 2.2 = 13.2
Right-endpoint estimate: Use the right endpoints of each subinterval to approximate the area. The right endpoints are at x = 2, 3, 4, and 5 with corresponding y-values of 3.5, 2.5, 2.2, and 2. The width of each subinterval is 1. Therefore, the right-endpoint estimate is:
A = 1*(3.5) + 1*(2.5) + 1*(2.2) + 1*(2) = 3.5 + 2.5 + 2.2 + 2 = 10.2
Trapezoidal estimate: Use the trapezoidal rule to approximate the area. The formula for the trapezoidal rule is:
A = (1/2) * (b-a) * (f(a) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(b))
Using the points (1, 5), (2, 3.5), (3, 2.5), (4, 2.2), and (5, 2), the trapezoidal estimate is:
A = (1/2) * (1) * (5 + 2*3.5 + 2*2.5 + 2*2.2 + 2) = (1/2) * (5 + 7 + 5 + 4.4 + 2) = (1/2) * 23.4 = 11.7
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