(a) Estimate the area under the graph of f(x) = 2/x from x = 1 to x = 2 using four approximating rectangles and right endpoints. (Round your answer to four decimal places.) Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?underestimateoverestimate    (b) Repeat part (a) using left endpoints. (Round your answer to four decimal places.

Estimate the area under the graph of f(x) = 2/x from x = 1 to x = 2 using four approximating rectangles and right endpoints. (Round your answer to four decimal places.)

Select the first function, y = 0.2x2, and set the interval to [−5, 0].(a) With 5 subintervals, is using rectangles with left endpoints an underestimate or an overestimate?underestimateoverestimate    How can you tell from the graph?The rectangles do not cover all the area under the function.The rectangles cover more area than just under the function.    It is difficult to determine whether it is an overestimate or underestimate just by looking at the graph.With 5 subintervals, is using rectangles with right endpoints an underestimate or an overestimate?underestimateoverestimate    (b) If you change to using midpoints, is the approximation an underestimate or an overestimate?underestimateoverestimate    How can you tell from the graph?The rectangles do not cover all the area under the function.The rectangles cover more area than just under the function.    It is difficult to determine whether it is an overestimate or underestimate just by looking at the graph.(c) Which of these three methods gives the best estimate?left endpoint approximationright endpoint approximation    midpoint approximation(d) If you increase the number of subintervals, will that change any of your answers to parts (a), (b), or (c)? Explain.Yes. If the number of intervals is increased the rectangles for the left endpoints will be below the function and the rectangles for right endpoints will be above the function.No, because the function is strictly decreasing, the rectangles for the left endpoints are always above the function and the rectangles for the right endpoints are always under the function.

Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval.f(x) = 2x + 9,   [0, 2],   4 rectangles

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.

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.