Problem

The function 𝑓(𝑥) = 𝑥² represents the area, in square units, between the graph of 𝑓 and the 𝑥-axis on the interval [0, 3]. The region is divided into three equal subintervals. Find the sum of the areas of the trapezoids that approximate the region, using the Midpoint Rule.

Answer

To find the sum of the areas of the trapezoids using the Midpoint Rule, we need to divide the interval [0, 3] into three equal subintervals and calculate the area under the curve for each subinterval.

Step 1: Find the width of each subinterval. Since the interval [0, 3] is divided into three equal subintervals, the width of each subinterval is (3-0)/3 = 1.

Step 2: Find the midpoints of each subinterval. To find the midpoints, we add half of the width to the left endpoint of each subinterval. The midpoints are:

• For the first subinterval [0, 1], the midpoint is (0 + 1/2) = 1/2.
• For the second subinterval (1, 2], the midpoint is (1 + 1 + 1/2) = 3/2.
• For the third subinterval (2, 3], the midpoint is (2 + 1 + 1/2) = 5/2.

Step 3: Calculate the height of each trapezoid. To find the height of each trapezoid, we evaluate the function 𝑓(𝑥) = 𝑥² at the midpoints calculated in step 2. The heights are:

• For the first trapezoid with midpoint 1/2, the height is 𝑓(1/2) = (1/2)² = 1/4.
• For the second trapezoid with midpoint 3/2, the height is 𝑓(3/2) = (3/2)² = 9/4.
• For the third trapezoid with midpoint 5/2, the height is 𝑓(5/2) = (5/2)² = 25/4.

Step 4: Calculate the area of each trapezoid. The area of each trapezoid is given by the formula: Area = (width/2) * (height1 + height2), where height1 and height2 are the heights of the trapezoid.

• For the first trapezoid, the area is (1/2) * (1/4 + 1/4) = 1/4.
• For the second trapezoid, the area is (1/2) * (9/4 + 9/4) = 9/4.
• For the third trapezoid, the area is (1/2) * (25/4 + 25/4) = 25/4.

Step 5: Sum the areas of the trapezoids. To find the sum of the areas of the trapezoids, we add the areas of all three trapezoids: Sum of areas = 1/4 + 9/4 + 25/4 = 35/4.

Therefore, the sum of the areas of the trapezoids that approximate the region, using the Midpoint Rule, is 35/4 square units.