Let f(x) = sin(x) + 1. Use a midpoint sum with 2 equal sub-intervals to estimate the area between f and the x-axis on [0, 2π].
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.
Example 2
Let g(x) = x3on the interval [0, 12]. Use a midpoint sum with 4 equal sub-intervals to estimate the area between g and the x-axis on this interval.
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.
Example 3
Partial values of the function h are given in the table below. Use a midpoint sum with 3 intervals to estimate the area between h and the x-axis on [-2, 4] or explain why you can't.
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.
Example 4
Partial values of the function f(x) are given in the table below. Use a midpoint sum with 4 equal sub-intervals to estimate the area between f and the x-axis on[0, 1], or explain why you can't.
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.
so the rectangle has height
so the rectangle has height
