Four rectangles are used to estimate the area under the curve, y=x3 , from x = 0 to x = 12. Using the left-hand rectangular approximation method (LRAM), what is the approximate area under the curve for the given interval? (1 point) Responses 972 972 2,700 2,700 2,916 2,916 5,022 5,022 8,100
To use the left-hand rectangular approximation method (LRAM), we divide the interval from x = 0 to x = 12 into four equal subintervals. The width of each subinterval will be 12/4 = 3.
To find the height of each rectangle, we evaluate the function y = x^3 at the left endpoint of each subinterval.
For the first rectangle, when x = 0, y = 0^3 = 0.
For the second rectangle, when x = 3, y = (3)^3 = 27.
For the third rectangle, when x = 6, y = (6)^3 = 216.
For the fourth rectangle, when x = 9, y = (9)^3 = 729.
Now we can find the area of each rectangle:
First rectangle: width = 3, height = 0, so area = 3*0 = 0.
Second rectangle: width = 3, height = 27, so area = 3*27 = 81.
Third rectangle: width = 3, height = 216, so area = 3*216 = 648.
Fourth rectangle: width = 3, height = 729, so area = 3*729 = 2187.
To find the approximate area under the curve, we sum up the areas of the rectangles:
0 + 81 + 648 + 2187 = 2916.
Therefore, the approximate area under the curve from x = 0 to x = 12, using the left-hand rectangular approximation method, is 2,916.