10 POINTS!!! FULL ANSWER IN STEP BY STEP FORMAT!!
2 answers:
It is convenient to start by understanding what the curve looks like in the region of interest. A graph can help. The limits of the sum will be from x=0 to x=3. We want each area in the sum to have the same width, so that width will be 3/3 = 1. That is, the area of each of the summands will be its height multiplied by 1, its width. In short, we can obtain the required sum by simply adding the height of the function at the appropriate points. Note that this process is eased immensely by having a table of values of the function.
a) The left end of each interval is where x ∈ {0, 1, 2}. The area is then
f(0) +f(1) +f(2) = 3 + 4 + 3 = 10
b) The right end of each interval is where x ∈ {1, 2, 3}. The area is then
f(1) +f(2) +f(3) = 4 + 3 + 0 = 7
c) The middle of each interval is where x ∈ {0.5, 1.5, 2.5}. The area is then
f(0.5) +f(1.5) +f(2.5) = 3.75 + 3.75 + 1.75 = 9.25
d) The trapezoidal rule averages the left and right ends of each interval. We can obtain the same result by averaging the Left Sum and the Right Sum.
(10 + 7)/2 = 8.5
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For comparison, the actual area under the curve in the first quadrant is 9.00.
a) The left end of each interval is where x ∈ {0, 1, 2}. The area is then
f(0) +f(1) +f(2) = 3 + 4 + 3 = 10
b) The right end of each interval is where x ∈ {1, 2, 3}. The area is then
f(1) +f(2) +f(3) = 4 + 3 + 0 = 7
c) The middle of each interval is where x ∈ {0.5, 1.5, 2.5}. The area is then
f(0.5) +f(1.5) +f(2.5) = 3.75 + 3.75 + 1.75 = 9.25
d) The trapezoidal rule averages the left and right ends of each interval. We can obtain the same result by averaging the Left Sum and the Right Sum.
(10 + 7)/2 = 8.5
_____
For comparison, the actual area under the curve in the first quadrant is 9.00.
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