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).![](https://media1.shmoop.com/images/calculus/calc_definiteint_lhs_latek_1.png)
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, 0.5]:
The left endpoint of this sub-interval is 0.
The height of this rectangle is
f (0) = 02 + 1 = 1
so the area is
(height) ⋅ (width) = 1 ⋅ (0.5) = 0.5
Sub-interval [0.5, 1]:
The left endpoint of this sub-interval is 0.5. The height of this rectangle is
f (0.5) = (0.5)2 + 1 = 1.25
so the area is
(height) ⋅ (width) = 1.25 ⋅ (0.5) = 0.625
Sub-interval [1, 1.5]. The left endpoint of this interval is 1. The height of this rectangle is
f (1) = 12 + 1 = 2
so the area is
(height) ⋅ (width) = 2 ⋅ (0.5) = 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 ⋅ (0.5) = 1.625
Sub-interval [2, 2.5]:
The height of this rectangle is
f (2) = 22 + 1 = 5
so the area is
(height) ⋅ (width) = 5 ⋅ (0.5) = 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) = 7.25 ⋅ (0.5) = 3.625.
Sub-interval [3, 3.5]. The height of this rectangle is
f (3) = 32 + 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) = 13.25 ⋅ (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.