# Definite Integrals

### Example 1

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

*x*+ 1 and the x-axis on the interval [0,4].

^{2}- 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 (b) and (c) 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.\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?

### 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 in (a) and (b) 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 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]?

### 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 in (a) and (b) 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].