# Calculus 1 : Functions

## Example Questions

### Example Question #11 : Midpoint Riemann Sums

Solve the integral

using the midpoint Riemann sum approximation with  subintervals.

Explanation:

Midpoint Riemann sum approximations are solved using the formula

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

For this problem, .

The approximate value at each midpoint is below.

The sum of all the approximate midpoints values is , therefore

### Example Question #12 : How To Find Midpoint Riemann Sums

Let

What is the Midpoint Riemann Sum on the interval  divided into four sub-intervals?

Explanation:

The interval  divided into four sub-intervals gives rectangles with vertices of the bases at

For the Midpoint Riemann sum, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or f(1), f(3), f(5), and f(7).

Because each interval has width 2, the approximated Midpoint Riemann Sum is

### Example Question #13 : How To Find Midpoint Riemann Sums

Using midpoint Riemann sum, approximate the area under the curve of  from  using 4 rectangular partitions.

Explanation:

Four rectangles from  to  would give us a .

Thus, there is a rectangle from  to , a rectangle from  to , a rectangle from  to , and a rectangle from  to .

To find the area, evaluate the function at each midpoint of all the rectangles to get the height, then multiply it by the width of the rectangle and sum it all together.

Hence, evaluate  at  and :

### Example Question #1 : Numerical Integration

Solve the integral

using the trapezoidal approximation with  subintervals.

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.

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.

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 #4 : Trapezoidal Rule

Solve the integral

using the trapezoidal approximation with  subintervals.

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 #11 : Functions

Solve the integral

using Simpson's rule with  subintervals.

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.

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 #16 : How To Find Midpoint Riemann Sums

Solve the integral

using Simpson's rule with  subintervals.

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