Six rectangles are used to estimate the area under the curve, y=−x+5 , from x = 0 to x = 3. Using the midpoint rectangular approximation method (MRAM), what is the approximate area under the curve for the given interval? (1 point) Responses 9.75 9.75 10.5 10.5 11.25 11.25 21 21 22.5 22.5
To estimate the area under the curve using MRAM, we divide the interval from x=0 to x=3 into six equal subintervals.
The width of each subinterval is: Δx = (3-0)/6 = 0.5
To approximate the area under the curve, we will use the rectangle's height at the midpoint of each subinterval.
The midpoint of each subinterval is given by: x = (x_i + x_(i+1))/2, where i represents the index of the subinterval.
For the first subinterval (i=0), the midpoint is x = (0 + 0.5)/2 = 0.25
The height of the rectangle at this midpoint is y = -(0.25) + 5 = 4.75
For the second subinterval (i=1), the midpoint is x = (0.5 + 1)/2 = 0.75
The height of the rectangle at this midpoint is y = -(0.75) + 5 = 4.25
For the third subinterval (i=2), the midpoint is x = (1 + 1.5)/2 = 1.25
The height of the rectangle at this midpoint is y = -(1.25) + 5 = 3.75
For the fourth subinterval (i=3), the midpoint is x = (1.5 + 2)/2 = 1.75
The height of the rectangle at this midpoint is y = -(1.75) + 5 = 3.25
For the fifth subinterval (i=4), the midpoint is x = (2 + 2.5)/2 = 2.25
The height of the rectangle at this midpoint is y = -(2.25) + 5 = 2.75
For the sixth subinterval (i=5), the midpoint is x = (2.5 + 3)/2 = 2.75
The height of the rectangle at this midpoint is y = -(2.75) + 5 = 2.25
Now, we can calculate the area of each rectangle and sum them up:
Area of rectangle 1 = Δx * y1 = 0.5 * 4.75 = 2.375
Area of rectangle 2 = Δx * y2 = 0.5 * 4.25 = 2.125
Area of rectangle 3 = Δx * y3 = 0.5 * 3.75 = 1.875
Area of rectangle 4 = Δx * y4 = 0.5 * 3.25 = 1.625
Area of rectangle 5 = Δx * y5 = 0.5 * 2.75 = 1.375
Area of rectangle 6 = Δx * y6 = 0.5 * 2.25 = 1.125
Total area under the curve ≈ Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 + Area of rectangle 4 + Area of rectangle 5 + Area of rectangle 6
≈ 2.375 + 2.125 + 1.875 + 1.625 + 1.375 + 1.125
≈ 10.5
Therefore, the approximate area under the curve using MRAM for the given interval is 10.5.