Let *W* be the area between the graph of and the *x*-axis on the interval [1,4].

- Draw W.\item Use a Right-Hand Sum with 3 subintervals to approximate the area of W. Draw W and the rectangles used in this Right-Hand Sum on the same graph.
- Use a Right-Hand Sum with 6 subintervals to approximate the area of W. Draw W and the rectangles used in this Right-Hand Sum on the same graph.
- Are your approximations in parts (b) and (c) bigger or smaller than the actual area of W?

Answer

- Dividing the interval [1,4] into 3 sub-intervals of equal length gives us sub-intervals of length 1.

Sub-interval [1,2]:

The height of this rectangle is

so the area of the rectangle is

Sub-interval [2,3]:

The height of this rectangle is

so the area of the rectangle is

Sub-interval [3,4]:

The height of this rectangle is

so the area of the rectangle is

Adding the areas of these rectangles, we estimate that the area of *W* is

- Dividing the interval [1,4] into 6 sub-intervals of equal length gives us sub-intervals of length .

Sub-interval [1,1.5]:

The height of this rectangle is

so the area is

Sub-interval [1.5,2]:

The height of this rectangle is

so the area is

Sub-interval [2,2.5]:

The height of this rectangle is

so the area is

Sub-interval [2.5,3]:

The height of this rectangle is

so the area is

Sub-interval [3,3.5]:

The height of this rectangle is

so the area is

Sub-interval [3.5,4]:

The height of this rectangle is

so the area is

Adding the areas of all these rectangles, we estimate that the area of *W* is

- The approximations in (b) and (c) are both underestimates, since the rectangles do not cover the entire area of
*W*.