Some values of the decreasing function *g* are given in the table below:

- Use a right-hand sum with 3 sub-intervals to estimate the area between the graph of
*g* and the *x*-axis on the interval [-1,2].

- Use a right-hand sum with 2 sub-intervals to estimate the area between the graph of
*g* and the *x*-axis on the interval [-1,2].

- Are your answers in (a) and (b) over- or under-estimates for the actual area between the graph of g and the
*x*-axis on the interval [-1,2]?

Answer

- With 3 sub-intervals we split the original interval [-1,2] into sub-intervals of length 1, so each rectangle will have width 1.

On [-1,0] the rectangle has height *g* (0) = 20 and area 20(1) = 20.

On [0,1] the rectangle has height *g* (1) = 18 and area 18(1) = 18.

On [1,2] the rectangle has height *g* (2) = 3 and area 3(1) = 3.

Adding the areas of these rectangles, we estimate that the area between the graph of *g* ( *x* ) and the *x*-axis on [-1,2] is

20 + 18 + 3 = 41.

- With 2 sub-intervals we split the original interval [-1,2] into sub-intervals of length 1.5.On [-1,.5] the rectangle has height
*g* (.5) = 19 and width 1.5, so its area is

19(1.5) = 28.5.

On [.5,2] the rectangle has height *g* (2) = 3 and width 1.5, so its area is

3(1.5) = 4.5.

Adding the areas of these rectangles, we estimate that the area between the graph of *g* ( *x* ) and the *x*-axis on [-1,2] is

28.5 + 4.5 = 33

- The estimates in (a) and (b) are both underestimates. The function
*g* ( *x* ) and the rectangles from the estimate in (a) would look something like this:

The rectangles don't cover the entire area between *g* and the *x*-axis on [-1,2], so we get an underestimate of that area.