Question

The region R enclosed between the graph of the function y=x^2−4x+5 and the x-axis for 0≤x≤4 is partitioned into four subintervals. Determine the areas of the four rectangles using MRAM and then approximate the area of R. Round areas to the nearest hundredth. Show and explain your work.

The answer I got for this problem is 9 square units.

Answer 1

9 units²

Explanation:

f(x) = x² - 4x + 5

Four intervals, their midpoints and f(MP) for heights:

0≤x<1 --> 0.5 --> 3.25

1<x≤2 --> 1.5 --> 1.25

2<x≤3 --> 2.5 --> 1.25

3<x≤4 --> 3.5 --> 3.25

All four intervals have a width of 1 unit

Sum of areas = total area

= 1×3.25 + 1×1.25 + 1×1.25 + 1×3.25

= 9

Answer 2

9.00 square units

Explanation:

The width of the interval is 4 − 0 = 4. Divided by 4 equal subintervals, the width of each subinterval is 4/4 = 1.

The subintervals are:

0 ≤ x ≤ 1

1 ≤ x ≤ 2

2 ≤ x ≤ 3

3 ≤ x ≤ 4

MRAM is midpoint rectangular approximation method. So we use the midpoints of each interval to find the height of the rectangle:

f(0.5) = (0.5)² − 4(0.5) + 5 = 3.25

f(1.5) = (1.5)² − 4(1.5) + 5 = 1.25

f(2.5) = (2.5)² − 4(2.5) + 5 = 1.25

f(3.5) = (3.5)² − 4(3.5) + 5 = 3.25

So the total approximate area is:

A = 3.25 + 1.25 + 1.25 + 3.25

A = 9.00

Graph: desmos.com/calculator/x8dcibqszo