# Definite Integrals

### Example 1

Let *R* be the area 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 Left-Hand Sum with 8 sub-intervals to approximate the area of*R*.

- Use the Left-Hand sum with 8 sub-intervals to approximate the area of
*R*(you might want a calculator).

### Example 2

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

- Draw
*S*.

- Use a Left-Hand Sum with 2 subintervals to approximate the area of
*S*. Draw*S*and the rectangles used in this Left-Hand Sum on the same graph.

- Use a Left-Hand Sum with 5 subintervals to approximate the area of
*S*. Draw*S*and the rectangles used in this Left-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*.

- Use a Left-Hand Sum with 3 subintervals to approximate the area of
*W*. Draw*W*and the rectangles used in this Left-Hand Sum on the same graph.

- Use a Left-Hand Sum with 6 subintervals to approximate the area of
*W*. Draw*W*and the rectangles used in this Left-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 left-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 left-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 left-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 left-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
*f*(*x*) =*x*^{2}+ 6*x*+ 9. Use a left-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 7

- Let
*f*(*x*) = -*x*^{2}+ 2*x*+ 8. Use a left-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 8

Let *g* be a function with values given by the table below.

Use a left-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 9

Let *h* be a function with values given by the table below. Use a left-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 10

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

### Example 11

Use a left-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].

### Example 12

Use a left-hand sum with the sub-intervals indicated by the data in the table to estimate the area between the graph of *g* and the *x*-axis on the interval [0,5π].

### Example 13

Use a left-hand sum to estimate the area between the graph of *h* and the *x*-axis on the interval [2,7].