(c) Estimate 9f(x) dx3 using three equal subintervals with midpoints.M3 =

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.

EXAMPLE 5 Use the Midpoint Rule with n = 5 to approximate the following integral.43x dx2SOLUTION The endpoints of the subintervals are 2, 2.4, 2.8, 3.2, 3.6, and 4, so the midpoints are 2.2, 2.6, 3, 3.4, and . The width of the subintervals is Δx = (4 − 2)/5 = , so the Midpoint Rule gives43x dx2≈ Δx[f(2.2) + f(2.6) + f(3) + f(3.4) + f(3.8)] =  0.432.2 + 32.6 + 33 + 33.4 + 33.8≈ . (Round your answer to four decimal places.)Since f(x) = 3x > 0 for 2 ≤ x ≤ 4, the integral represents an area, and the approximation given by the Midpoint Rule is the sum of the areas of the rectangles shown in the figure.

A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for 30f(x) dx.10lower estimate     upper estimate

A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for 30f(x) dx.10lower estimate     upper estimate     x 10 14 18 22 26 30f(x)−11−5−1257

Select the fourth function, y = 1x2 + 1, and set the interval to [−3, 2].(a) Find the approximate net area for 5 subintervals using left-endpoint rectangles.Find the approximate net area for 5 subintervals using right-endpoint rectangles.Find the approximate net area for 5 subintervals using trapezoids.(b) Compute the average of the two rectangle approximations from part (a) and compare this to the trapezoidal estimate. What do you notice?The average of the left and right endpoint approximations is equal to twice the trapezoid approximation.The average of the left and right endpoint approximations is equal to a fourth of the trapezoid approximation.    The average of the left and right endpoint approximations is equal to four times the trapezoid approximation.The average of the left and right endpoint approximations is equal to the trapezoid approximation.The average of the left and right endpoint approximations is equal to half the trapezoid approximation.(c) For 10 subintervals, which is more accurate, using trapezoids or rectangles with midpoints?Using trapezoids is more accurate.Using rectangles with midpoints is more accurate.    The methods are equally accurate.How do the errors compare?The error using trapezoids is about half the midpoint approximation error.The error using trapezoids is about twice the midpoint approximation error.    The error using trapezoids is equal to the midpoint approximation error.The error using trapezoids is about a fourth of the midpoint approximation error.The error using trapezoids is about four times the midpoint approximation error.(d) Click the Simpson button and use Simpson's Rule to approximate the net area with 10 subintervals. Is this more accurate than the Trapezoidal Rule's estimate?YesNo    (e) Which is more accurate, Simpson's Rule with 10 subintervals or the Trapezoidal Rule with 50 subintervals?Simpson's Rule with 10 subintervalsTrapezoidal Rule with 50 subintervals    By how much do these estimates differ? (Round your answer to five decimal places.)(f) Of the available choices, how many subintervals are needed for the midpoint approximation to be more accurate than Simpson's Rule with 10 subintervals?The midpoint approximation with 15 subintervals is more accurate than Simpson's Rule with 10 subintervals.The midpoint approximation with 26 subintervals is more accurate than Simpson's Rule with 10 subintervals.    The midpoint approximation with 38 subintervals is more accurate than Simpson's Rule with 10 subintervals.The midpoint approximation with 50 subintervals is more accurate than Simpson's Rule with 10 subintervals.Simpson's Rule with 10 subintervals is still more accurate than the midpoint approximation with 50 subintervals.