If the area under the curve of f(x) = x2 + 2 from x = 1 to x = 6 is estimated using five approximating rectangles and right endpoints, will the estimate be an underestimate or overestimate?
Underestimate
Overestimate
The area will be exact
The area cannot be estimated with just five rectangles <-MY ANSWERR
Obviously it will be an overestimate. However, it is still an estimate. I cannot agree with your answer. Estimates are estimates, not exact. One could estimate it with just one rectangle, as inaccurate as it would be
To determine whether the estimate will be an underestimate or an overestimate, we need to consider the behavior of the function f(x) = x^2 + 2 from x = 1 to x = 6 using right endpoints and approximating rectangles.
One way to estimate the area under a curve is by using Riemann sums, where the area is approximated by dividing the region into rectangles and summing up their areas. The choice of right endpoints means that the right side of each rectangle will touch the curve. In this case, we are using five approximating rectangles.
To estimate the area using right endpoints and rectangles, we need to calculate the width and height of each rectangle. The width of each rectangle will be given by Δx = (6 - 1)/5 = 1. The height of each rectangle will be given by the value of f(x) at the right endpoint of each interval.
Given that the function f(x) = x^2 + 2 is increasing on the interval [1, 6], using right endpoints means that the height of each rectangle will be determined by the function value at the right endpoint of each interval. In this case, the right endpoints are x = 2, 3, 4, 5, and 6.
Calculating the height of each rectangle:
f(2) = 2^2 + 2 = 6
f(3) = 3^2 + 2 = 11
f(4) = 4^2 + 2 = 18
f(5) = 5^2 + 2 = 27
f(6) = 6^2 + 2 = 38
Calculating the area of each rectangle:
Rectangle 1: A1 = (1)(6) = 6
Rectangle 2: A2 = (1)(11) = 11
Rectangle 3: A3 = (1)(18) = 18
Rectangle 4: A4 = (1)(27) = 27
Rectangle 5: A5 = (1)(38) = 38
Adding up the areas of the rectangles:
Total area ≈ A1 + A2 + A3 + A4 + A5
≈ 6 + 11 + 18 + 27 + 38
≈ 100
Therefore, the estimate for the area under the curve of f(x) = x^2 + 2 from x = 1 to x = 6 using five approximating rectangles and right endpoints is 100 square units.
Comparing the estimate to the actual area under the curve, we can conclude that the estimate using right endpoints and five rectangles will be an overestimate.