# Calculus 2 : Integrals

## Example Questions

### Example Question #1 : Riemann Sum: Midpoint Evaluation

Approximate

using the midpoint rule with . Round your answer to three decimal places.

Explanation:

The interval  is 4 units in width; the interval is divided evenly into four subintervals  units in width, with their midpoints shown:

The midpoint rule requires us to calculate:

where  and

Evaluate  for each of :

So

### Example Question #1 : Riemann Sum: Midpoint Evaluation

Approximate

using the midpoint rule with . Round your answer to three decimal places.

Cannot be determined

Explanation:

The interval  is 1 unit in width; the interval is divided evenly into five subintervals  units in width, with their midpoints shown:

The midpoint rule requires us to calculate:

where  and

Evaluate  for each of :

### Example Question #41 : Integrals

Approximate

using the trapezoidal rule with . Round your answer to three decimal places.

Explanation:

The interval  is 1 unit in width; the interval is divided evenly into five subintervals  units in width. They are

.

The trapezoidal rule approximates the area of the given integral  by evaluating

,

where

and

.

So

### Example Question #1 : Trapezoidal Sums

Approximate

using the trapezoidal rule with . Round your answer to three decimal places.

Explanation:

The interval  is  units in width; the interval is divided evenly into four subintervals  units in width. They are

.

The trapezoidal rule approximates the area of the given integral  by evaluating

,

where

,

,

and

.

So

### Example Question #1 : Trapezoidal Sums

Approximate

using the trapezoidal rule with . Round your estimate to three decimal places.

Explanation:

The interval  is 4 units in width; the interval is divided evenly into four subintervals  units in width - they are .

The trapezoidal rule approximates the area of the given integral  by evaluating

,

where , and

.

### Example Question #31 : Numerical Approximations To Definite Integrals

Approximate

using the midpoint rule with . Round your answer to three decimal places.

None of the other choices are correct.

Explanation:

The interval  is  units in width; the interval is divided evenly into five subintervals  units in width, with their midpoints shown:

The midpoint rule requires us to calculate:

where  and

Evaluate  for each of :

Since ,

we can approximate  as

.

### Example Question #41 : Integrals

Using Simpson's parabolic rule with , give an approximation of

.

Round your approximation to the nearest thousandth.

Explanation:

The interval  is divided into four subintervals of width  by the numbers

Simpson's parabolic rule tells us that we can approximate

with  using the formula

,

where

,

and

.

Therefore,

.

Evaluate  at each of these values of :

Substitute:

### Example Question #41 : Integrals

Using Simpson's parabolic rule with , give an approximation of

.

Round your approximation to the nearest thousandth.

Explanation:

The interval  is divided into four subintervals of width  by the numbers .

Simpson's parabolic rule tells us that we can approximate

with  using the formula

,

where

,

and

.

Therefore,

Evaluate  at each of these values of :

Substitute:

### Example Question #1791 : Calculus Ii

Using Simpson's parabolic rule with , give, to the nearest thousandth, an approximation of

.

Explanation:

The interval  is divided into four subintervals of width  by the numbers .

Simpson's parabolic rule tells us that we can approximate

with  using the formula

,

where

,

and

.

Therefore,

Evaluate  for each of these values, leaving the results in logarithmic form for the time being:

So

### Example Question #1 : Riemann Sums

Find the Left Riemann sum of the function

on the interval  divided into four sub-intervals.