Question

F(x)= x² 5 a=-2, b=9, and n=5 estimate the area of the region using left and right endpoints.

Options:
a) 28.8 and 48.6
b) 57.6 and 39.2
c) 44.8 and 36.2
d) 33.6 and 52.7

Answer 1

To estimate the area of the region using left and right endpoints, we divide the interval into subintervals and evaluate the function at the endpoints of each subinterval. The estimated area using the left endpoints is 77.2 square units and the estimated area using the right endpoints is 154.2 square units.

Step-by-step explanation:

To estimate the area of the region using left and right endpoints, we can use the Riemann sum method. Since n=5, we divide the interval from a=-2 to b=9 into 5 subintervals of equal width. The width of each subinterval is (b-a)/n = (9-(-2))/5 = 11/5 = 2.2.

To estimate the area using the left endpoints, we evaluate the function at the left endpoints of each subinterval and sum the results. F(-2) = (-2)^2 = 4, F(-2+2.2) = 0.2^2 = 0.04, F(-2+2.2*2) = 2.4^2 = 5.76, F(-2+2.2*3) = 4.6^2 = 21.16, F(-2+2.2*4) = 6.8^2 = 46.24.

Adding these values together, we get the estimated area using the left endpoints as: 4+0.04+5.76+21.16+46.24 = 77.2. Therefore, the estimated area using the left endpoints is 77.2 square units.

Similarly, to estimate the area using the right endpoints, we evaluate the function at the right endpoints of each subinterval and sum the results. F(-2+2.2) = 0.2^2 = 0.04, F(-2+2.2*2) = 2.4^2 = 5.76, F(-2+2.2*3) = 4.6^2 = 21.16, F(-2+2.2*4) = 6.8^2 = 46.24, F(9) = 9^2 = 81.

Adding these values together, we get the estimated area using the right endpoints as: 0.04+5.76+21.16+46.24+81 = 154.2. Therefore, the estimated area using the right endpoints is 154.2 square units.