# Definite Integrals

### Example 1

Let *f* (*x*) = *x*^{2}. Calculate the following quantities.

- (
*f*(12)-*f*(0))(12)

- (
*f*(12)-*f*(0))(6)

- (
*f*(12)-*f*(0))(4)

- (
*f*(12)-*f*(0))(3)

- (
*f*(12)-*f*(0))(1)

### Example 2

Let *f* (*x*) = *x*^{2} on the interval [0,12]. How many rectangles must we use to guarantee that the right- and left-hand sums are within 10
of each other?

### Example 3

Let *f* (*x*) = 4*x* + 6 on the interval [2,8]. How many rectangles must we use to ensure that the left-hand sum is within 0.25 of the exact area between *f* and the *x*-axis on this interval?

### Example 4

Let *f* (*x*) = *x*^{4} on the interval [-2,-1]. For what values of *n* are *LHS*(*n*) and *RHS*(*n*) within 0.5 of each other on this interval?

### Example 5

Let *f* (*x*) = cos(*x*) on the interval How large must *n* be to guarantee that *LHS*(*n*) is within 0.3 of the exact area between *f* and the *x*-axis on this interval?