Use the table of values to estimate
⌠6
⎮ f(x)⋅dx
⌡0
Use three sub intervals and the (a) left endpoints, (b) right endpoints, and (c) midpoints. If f is an increasing function, how does each estimate compare with the actual value? Explain your reasoning.
x:
{ 0, 1, 2, 3, 4, 5, 6}
f(x):
{-6, 0, 8,18,30,50,80}
To estimate the value of the integral ⌠6 ⎮ f(x)⋅dx ⌡0 using three sub-intervals and the (a) left endpoints, (b) right endpoints, and (c) midpoints, we first need to calculate the width of each sub-interval. In this case, the total interval is from x=0 to x=6, so the width of each sub-interval is 6/3 = 2.
(a) Using the left endpoints of the sub-intervals:
- The left endpoint of the first sub-interval is x=0, where f(x)=-6.
- The left endpoint of the second sub-interval is x=2, where f(x)=8.
- The left endpoint of the third sub-interval is x=4, where f(x)=30.
Now, we can calculate the approximate value of the integral by multiplying each value of f(x) with the width of the sub-interval and summing them up:
Approximation using left endpoints = (-6 * 2) + (8 * 2) + (30 * 2) = -12 + 16 + 60 = 64.
(b) Using the right endpoints of the sub-intervals:
- The right endpoint of the first sub-interval is x=2, where f(x)=8.
- The right endpoint of the second sub-interval is x=4, where f(x)=30.
- The right endpoint of the third sub-interval is x=6, where f(x)=80.
Approximation using right endpoints = (8 * 2) + (30 * 2) + (80 * 2) = 16 + 60 + 160 = 236.
(c) Using the midpoints of the sub-intervals:
- The midpoint of the first sub-interval is x=1, where f(x)=0.
- The midpoint of the second sub-interval is x=3, where f(x)=18.
- The midpoint of the third sub-interval is x=5, where f(x)=50.
Approximation using midpoints = (0 * 2) + (18 * 2) + (50 * 2) = 0 + 36 + 100 = 136.
Given that f is an increasing function, each estimate can be compared with the actual value of the integral.
To compare the estimates with the actual value, we need to know the actual value of the integral ⌠6 ⎮ f(x)⋅dx ⌡0. Without this information, we cannot determine how accurate or close the estimates using the left endpoints, right endpoints, and midpoints are.
However, we can provide general reasoning based on the properties of increasing functions. Since f is an increasing function, the estimate using the left endpoints is likely to be an underestimate of the actual value because the left endpoints tend to be lower than the corresponding function values. On the other hand, the estimate using the right endpoints is likely to be an overestimate of the actual value since the right endpoints tend to be higher than the corresponding function values. The estimate using the midpoints may be more accurate since they more closely represent the average behavior of the function within each sub-interval.
In conclusion, without knowing the actual value of the integral, we cannot provide concrete comparisons between the estimates and the actual value. However, based on the properties of increasing functions, the estimation using the left endpoints is likely to be a lower bound, using the right endpoints is likely to be an upper bound, and using the midpoints may provide a reasonable estimate.