# AP Calculus AB : Trapezoidal sums

## Example Questions

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### Example Question #41 : Trapezoidal Sums

Using the Trapezoidal Rule with four subintervals, approximate to four decimal places:

Explanation:

The Trapezoidal Rule states that a definite integral

can be approximated by computing the expression

where  for

Since we are dividing  into four subintervals, set

; the expression becomes

where

### Example Question #42 : Trapezoidal Sums

Solve in the integral by method of trapezoidal sums

Explanation:

To use the trapezoidal rule, we apply the following formula:

Using the integral from our problem statement, we get

### Example Question #43 : Trapezoidal Sums

Evaluate the integral using the trapezoidal sums method:

Explanation:

To solve the integral, we will use the formula for the trapezoidal sums method:

Using the integral from the problem statement, we get

Simplifying, we end up with

### Example Question #44 : Trapezoidal Sums

Evaluate using the trapezoidal approximation:

Explanation:

The trapezoidal approximation of a definite integral is given by

For our integral, we get

### Example Question #45 : Trapezoidal Sums

Use the method of trapezoidal sums to approximate the integral

Explanation:

To approximate the integral using trapezoidal sums, we use the following formula:

Using the integral from the problem statement, we get

### Example Question #46 : Trapezoidal Sums

Using the method of trapezoidal sums, evaluate the following integral

Explanation:

To find the value of the integral, we use the following formula

### Example Question #47 : Trapezoidal Sums

Using the method of trapezoidal sums, evaluate the following integral

Explanation:

To use the method of trapezoidal sums, we follow the definition

Using the information from the problem statement, we get

### Example Question #48 : Trapezoidal Sums

Use the trapezoidal approximation to solve the definite integral, and find the difference between it and the actual integral:

Explanation:

The trapezoidal approximation to definite integrals is given by

Using this formula for our integral, we get

Actually integrating, we get

The rule used for integration is

The difference between the approximation and the actual answer is

### Example Question #49 : Trapezoidal Sums

Approximate the value of  using a trapezoidal sum with step size . How far away is this approximation from the actual value of the integral above?

6

5

10

2

4

4

Explanation:

Trapezoidal sums are found by creating trapezoids whose left and right end points are on the specified function, and whose widths are the step size. We then sum up their areas by remembering that the area of a trapezoid is the base times the average of the heights.

Thus, the calculation of the trapezoidal sum for this example would be

A more simplified version would be given by

Which evaluates to

The actual answer is found by evaluating the definite integral given, which would just be given by

The difference between the approximation and the the true answer is thus

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