# Right-Hand Sum Exercises

### Example 1

Let *R* be the region between the graph of *f*(* x*) =

*x*

^{2}+ 1 and the

*x*-axis on the interval [0, 4].

- Draw
*R*and the 8 rectangles that result from using a right-hand sum with 8 sub-intervals to approximate the area of*R*.

- Use a Right-Hand Sum with 8 sub-intervals to approximate the area of
*R*.

- Is your approximation an under-estimate or an over-estimate to the actual area of
*R*?

### Example 2

Let *S* be the area between the graph of *y* = *f *(*x*) = 2^{x} and the *x*-axis on the interval [1, 6].

- Draw S.
- Use a Right-Hand Sum with 2 subintervals to approximate the area of S. Draw S and the rectangles used in this Right-Hand Sum on the same graph.
- Use a Right-Hand Sum with 5 subintervals to approximate the area of S. Draw S and the rectangles used in this Right-Hand Sum on the same graph.
- Are your approximations in parts bigger or smaller than the actual area of S?

### Example 3

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

- Draw
*W*. - 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 bigger or smaller than the actual area of
*W*?

### Example 4

The table below shows some values of the increasing function *f*(*x*).

- Use a right-hand sum with one sub-interval to estimate the area between the graph of
*f*and the*x*-axis on the interval [2, 8].

- Use a right-hand sum with three sub-intervals to estimate the area between the graph of
*f*and the*x*-axis on the interval [2, 8].

- Are your answers over- or under-estimates of the actual area between the graph of
*f*and the*x*-axis on the interval [2, 8]?

### Example 5

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 over- or under-estimates for the actual area between the graph of g and the
*x*-axis on the interval [-1, 2]?

### Example 6

- Let
*W*be the region between the graph of*f*and the*x*-axis on the interval [-20, 20].

Use a right-hand sum with 4 sub-intervals to estimate the area of *W*.

### Example 7

- Let
*Z*be the region between the graph of*g*and the*x*-axis on the interval [-4, 0].

- Use a right-hand sum with 2 sub-intervals to estimate the area of
*Z*.

- Use a right-hand sum with 4 sub-intervals to estimate the area of
*Z*.

- Are your answers over- or under- estimates for the area of
*Z*?

### Example 8

Let *f*(*x*) = *x*^{2} + 6*x* + 9. Use a right-hand sum with 6 sub-intervals to estimate the area between the graph of *f* and the *x*-axis on the interval [-6,-3].

### Example 9

Let *f* (*x*) = -*x*^{2} + 2*x* + 8. Use a right-hand sum with 8 sub-intervals to estimate the area between the graph of *f* and the *x*-axis on the interval [0, 4].

### Example 10

Let *g* be a function with values given by 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 [0, 12].

### Example 11

Let *h* be a function with values given by the table below. Use a right-hand sum with 9 sub-intervals to estimate the area between the graph of *h* and the *x*-axis on the interval [-9, 9].

### Example 12

The function *f* (*x*) on the interval [0, 30] is graphed below. Use a right-hand sum with 3 sub-intervals to estimate the area between the graph of *f* and the *x*-axis on this interval.

### Example 13

Use a right-hand sum with the sub-intervals indicated by the data in the table to estimate the area between the graph of *f* and the *x*-axis on the interval [-10,1].