Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #1 : Riemann Sum: Midpoint Evaluation

Approximate

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

Possible Answers:

Correct answer:

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.

Possible Answers:

Cannot be determined

Correct answer:

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.

Possible Answers:

Correct answer:

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.

Possible Answers:

Correct answer:

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.

Possible Answers:

 

Correct answer:

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.

Possible Answers:

None of the other choices are correct.

Correct answer:

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.

Possible Answers:

Correct answer:

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.

Possible Answers:

Correct answer:

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 

.

 

Possible Answers:

Correct answer:

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.

Possible Answers:

Correct answer:

Explanation:

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

For the Left Riemann sum, we need to find the rectangle heights which values come from the left-most function value of each sub-interval, or f(0), f(2), f(4), and f(6).

Because each sub-interval has a width of 2, the Left Riemann sum is

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