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Example Questions
Example Question #13 : Riemann Sum: Midpoint Evaluation
Example Question #151 : Ap Calculus Bc
Example Question #43 : Integrals
Approximate
using the trapezoidal rule with . Round your answer to three decimal places.
The interval  is 1 unit in width; the interval is divided evenly into five subintervalsÂ
 units in width. They areÂ
.
The trapezoidal rule approximates the area of the given integral  by evaluatingÂ
,
whereÂ
andÂ
.
So
Example Question #152 : Ap Calculus Bc
Approximate
using the trapezoidal rule with . Round your answer to three decimal places.
The interval  is
 units in width; the interval is divided evenly into four subintervalsÂ
 units in width. They areÂ
.
The trapezoidal rule approximates the area of the given integral  by evaluatingÂ
,
whereÂ
,
,
andÂ
.
So
Â
Example Question #44 : Integrals
Approximate
using the trapezoidal rule with . Round your estimate to three decimal places.
Â
The interval  is 4 units in width; the interval is divided evenly into four subintervalsÂ
 units in width - they areÂ
.
The trapezoidal rule approximates the area of the given integral  by evaluatingÂ
,
where ,Â
, andÂ
.
Example Question #1 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation
Find the result:
Set . ThenÂ
, and by the chain rule,
By the fundamental theorem of Calculus, the above can be rewritten as
Example Question #2 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation
Evaluate :
Â
By the Fundamental Theorem of Calculus, we have that . Thus,
.Â
Example Question #1 : Fundamental Theorem Of Calculus
Evaluate  whenÂ
.
Via the Fundamental Theorem of Calculus, we know that, given a function,Â
.
Therefore .
Example Question #2 : Fundamental Theorem Of Calculus
Evaluate  whenÂ
.
Via the Fundamental Theorem of Calculus, we know that, given a function ,Â
. Therefore,Â
.
Example Question #2 : Fundamental Theorem Of Calculus
Suppose we have the function
What is the derivative, ?
We can view the function  as a function ofÂ
, as so
where .
We can find the derivative of  using the chain rule:
where  can be found using the fundamental theorem of calculus:
So we get
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