# GRE Subject Test: Math : Numerical Integration

## Example Questions

### Example Question #1 : Simpson's Rule

Solve the integral

using Simpson's rule with  subintervals.

Possible Answers:

Correct answer:

Explanation:

Simpson's rule is solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is  therefore

### Example Question #1 : Numerical Approximation

Solve the integral

using Simpson's rule with  subintervals.

Possible Answers:

Correct answer:

Explanation:

Simpson's rule is solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is  therefore

### Example Question #2 : Numerical Approximation

Solve the integral

using Simpson's rule with  subintervals.

Possible Answers:

Correct answer:

Explanation:

Simpson's rule is solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is  therefore

### Example Question #2 : Simpson's Rule

Solve the integral

using Simpson's rule with  subintervals.

Possible Answers:

Correct answer:

Explanation:

Simpson's rule is solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is  therefore

### Example Question #1 : Numerical Integration

Solve the integral

using the trapezoidal approximation with  subintervals.

Possible Answers:

Correct answer:

Explanation:

Trapezoidal approximations are solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is , therefore

### Example Question #2 : Trapezoidal Rule

Solve the integral

using the trapezoidal approximation with  subintervals.

Possible Answers:

Correct answer:

Explanation:

Trapezoidal approximations are solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is , therefore

### Example Question #1 : Numerical Integration

Solve the integral

using the trapezoidal approximation with  subintervals.

Possible Answers:

Correct answer:

Explanation:

Trapezoidal approximations are solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is , therefore

### Example Question #3 : Trapezoidal Rule

Solve the integral

using the trapezoidal approximation with  subintervals.

Possible Answers:

Correct answer:

Explanation:

Trapezoidal approximations are solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The value of each approximation term is below.

The sum of all the approximation terms is , therefore

### Example Question #2 : Trapezoidal Rule

Evaluate   using the Trapezoidal Rule, with n = 2.

Possible Answers:

Correct answer:

Explanation:

1) n = 2 indicates 2 equal subdivisions. In this case, they are from 0 to 1, and from 1 to 2.

2) Trapezoidal Rule is:

3) For n = 2:

4) Simplifying: