The table gives the values of a function obtained from an experiment. Use them to estimate 9f(x) dx3 using three equal subintervals with right endpoints, left endpoints, and midpoints.x 3 4 5 6 7 8 9f(x) −3.3 −2.2 −0.6 0.2 0.8 1.3 1.9(a) Estimate 9f(x) dx3 using three equal subintervals with right endpoints.
Question
The table gives the values of a function obtained from an experiment. Use them to estimate 9f(x) dx3 using three equal subintervals with right endpoints, left endpoints, and midpoints.x 3 4 5 6 7 8 9f(x) −3.3 −2.2 −0.6 0.2 0.8 1.3 1.9(a) Estimate 9f(x) dx3 using three equal subintervals with right endpoints.
Solution
To estimate ∫9f(x) dx from 3 to 9 using three equal subintervals with right endpoints, we first need to determine the width of each subinterval. The interval from 3 to 9 is 6 units wide, so each subinterval will be 6/3 = 2 units wide.
The right endpoints of these subintervals are x = 5, x = 7, and x = 9. We use the given function values at these points to estimate the integral.
The estimate is then given by the sum of the function values at these points times the width of the subintervals:
∫9f(x) dx ≈ 2[f(5) + f(7) + f(9)] = 2[(-0.6) + 0.8 + 1.9] = 2[2.1] = 4.2
So, the estimate of ∫9f(x) dx from 3 to 9 using three equal subintervals with right endpoints is 4.2.
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