Answer

To find the length of each sub-interval, we take the length of the original interval (4) and divide by the number of sub-intervals we want to chop it into (8).

- On each sub-interval, we go to the
**left endpoint** of that sub-interval and go up until we hit the function. The *y*-value of the function there is the height of the rectangle on that sub-interval.

- We need to find the area of each rectangle. Sub-interval [0,.5]:The left endpoint of this sub-interval is 0.The height of this rectangle is

*f* ( 0 ) = 0^{2} + 1 = 1

so the area is

(height) ⋅ (width) = 1 ⋅ (0.5) = .5

Sub-interval [.5,1]:

The left endpoint of this sub-interval is 0.5. The height of this rectangle is

*f* (.5) = .5^{2} + 1 = 1.25

so the area is

(height) ⋅ (width) = 1.25 ⋅ (.5) = .625

Sub-interval [1,1.5]. The left endpoint of this interval is 1. The height of this rectangle is

*f* (1) = 1^{2} + 1 = 2

so the area is

(height) ⋅ (width) = 2 ⋅ (.5) = 1

Sub-interval [1.5,2]:

The height of this rectangle is

*f* (1.5) = (1.5)^{2} + 1 = 3.25

so the area is

(height) ⋅ (width) = 3.25 ⋅ (.5) = 1.625

Sub-interval [2,2.5]:

The height of this rectangle is

*f* ( 2 ) = 2^{2} + 1 = 5

so the area is

(height) ⋅ (width) = 5 ⋅ (.5) = 2.5

Sub-interval [2.5,3]:

The height of this rectangle is

*f* (2.5) = (2.5)^{2} + 1 = 7.25

so the area is

(height) ⋅ (width) = 1 ⋅ (.5) = 3.625.

Sub-interval [3,3.5]. The height of this rectangle is

*f* (3) = 3^{2} + 1 = 10

so the area is

(height) ⋅ (width) = 10 ⋅ (0.5) = 5.

Sub-interval [3.5,4]. The left endpoint of this sub = interval is 3.5.

The height of this rectangle is

*f* (3.5) = (3.5)^{2} + 1 = 13.25

so the area is

(height) ⋅ (width) = 1 ⋅ (0.5) = 6.625

Adding up the areas of all 8 rectangles, we get

0.5 + 0.625 + 1 + 1.625 + 2.5 + 3.625 + 5 + 6.625 = 21.5.