Select the third function, y = 2 cos(x), and set the interval to [−4.02, 4.02].(a) With 10 rectangles using left endpoints, how many rectangles are contributing negative area values to the estimated net area?How many are positive?Is this the same as when using midpoints?YesNo    (b) What is the error when using midpoints with 10 subintervals? (Do not round your answer.)What is the ratio of the left-endpoint approximation error to the midpoint error? (Round your answer to two decimal places.)(c) For 20 subintervals, find the ratio of the error using left endpoints to the midpoint error. How does this compare to your answer in part (b)?The ratio is about the same as part (b).The ratio is significantly larger than part (b).    The ratio is significantly smaller than part (b).What about for 50 subintervals?The ratio is about the same as part (b).The ratio is significantly larger than part (b).    The ratio is significantly smaller than part (b).(d) Of the available choices, how many subintervals are required to give a midpoint approximation accurate to 2 decimal places?(e) Estimate an interval that would give a true net area of 0.[1.56, 3.14][−4.01, 0]    [0, 4.01][0, 3.14][−3.14, −1.57]

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.

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

(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.

Use Simpson's Rule with 10 subintervals to estimate the area under the graph of y = 0.2x2 for −4 ≤ x ≤ 2.