Estimating Area Under a Curve Using Midpoint Rectangles

The video tutorial explains how to estimate the area under the curve of the function f(x) = √(x+3) over the interval [-3, 3] using six rectangles and midpoints. It covers marking the interval, calculating the width of sub-intervals, sketching rectangles, and computing their areas using function values at midpoints. The final area approximation is calculated and rounded to four decimal places, resulting in approximately 9.8503 square units.

What is the function f(x) used in the problem?

• f(x) = sqrt(x) + 3

• f(x) = x + 3

• f(x) = sqrt(x + 3)

• f(x) = x^2 + 3

How many rectangles are used to approximate the area under the curve?

• 4

• 7

• 5

• 6

What is the width of each sub-interval?

• 0.5

• 1

• 1.5

• 2

What is the midpoint of the sub-interval from -3 to -2?

• -2.5

• -2

• -1.5

• -1

How is the height of each rectangle determined?

• Using the average of the endpoints

• Using the midpoint of the sub-interval

• Using the right endpoint of the sub-interval

• Using the left endpoint of the sub-interval

What is the function value at the midpoint of the sub-interval from 0 to 1?

• sqrt(0.5)

• sqrt(3.5)

• sqrt(1.5)

• sqrt(2.5)

What is the area of a rectangle with width 1 and height f(1.5)?

• sqrt(3.5)

• sqrt(1.5)

• sqrt(2.5)

• sqrt(4.5)

What is the sum of the function values used to estimate the area?

• Sum of f(-3) to f(3)

• Sum of f(-1.5) to f(1.5)

• Sum of f(-2.5) to f(2.5)

• Sum of f(-0.5) to f(0.5)

What is the estimated area under the curve using the midpoint method?

• 10.8503

• 11.8503

• 9.8503

• 8.7503

What is the interval over which the area is estimated?

• From -3 to 3

• From -1 to 1

• From -2 to 2

• From -4 to 4