Definite Integrals Exercises

Answer

Dividing [0, 2π] into 2 equal sub-intervals gets us the sub-intervals [0, π] and [π, 2π], each of which has width π.On [0, π] the midpoint is so the rectangle has height

and area

2(π) = 2π.

On [π, 2π] the midpoint is  so the rectangle has height

and area 0.

Adding the areas of the rectangles, we estimate the area between f and the x-axis on [0, 2π] to be 2π.

Answer

Dividing [0, 12] into 4 equal sub-intervals gets us sub-intervals of length 3.

On sub-interval [0, 3] we have

On sub-interval [3, 6] we have

On sub-interval [6, 9] we have

On sub-interval [9, 12] we have

Adding up the areas of these rectangles we get

10.125 + 273.375 + 1265.625 + 3472.875 = 5022.

Answer

Dividing [-2, 4] into 3 sub-intervals gets us sub-intervals [-2, 0], [0, 2], and [2, 4]. The midpoints of these sub-intervals are -1, 1, and 3. Since the table tells us what h is at these midpoints, we're able to use a midpoint sum.

On [-2, 0] we have

On [0, 2] we have

On [2, 4] we have

Adding the areas of the rectangles we get

12 + 4 + 0 = 16

as our estimate.

Answer

Dividing [0, 1] into 4 equal sub-intervals gets us the sub-intervals [0, 0.25], [0.25, 0.5], [0.5, 0.75], and [0.75, 1]. The midpoints of these sub-intervals are

0.125, 0.375, 0.625, and 0.875.

The table doesn't tell us what f is at these points, so we can't use the requested midpoint sum.