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Example Questions
Example Question #1 : Use Riemann Sums
The function has the following values on the interval
:
Approximate the integral of on this interval using right Riemann sums.
A Riemann sum integral approximation over an interval with
subintervals follows the form:
It is essentially a sum of rectangles, each with a base of length:
and variable heights:
, which depend on the function value at
.
In our case, we have function values over an interval:
Although the total interval has a length of .
Notice the points are equidistant. This distance is our subinterval and since we are using the right point of each interval, we disregard the first function value:
Example Question #4 : Integrating
Approximate
Using a right Riemann sum, if has the values:
A Riemann sum integral approximation over an interval with
subintervals follows the form:
It is essentially a sum of rectangles, each with a base of length:
and variable heights:
, which depend on the function value at
.
In our case, we have function values over an interval:
Although the total interval has a length of notice the points are equidistant. This distance is our subinterval:
. Since we are using the right point of each interval, we disregard the first function value:
Example Question #5 : Integrating
Approximate
Using a right Riemann sum, if has the values:
A Riemann sum integral approximation over an interval with
subintervals follows the form:
It is essentially a sum of rectangles, each with a base of length:
and variable heights:
, which depend on the function value at
.
In our case, we have function values over an interval:
Although the total interval has a length of notice the points are equidistant, with
intervals between. Each equidistant interval has length: 5
Since we are using the right point of each interval, we disregard the first function value:
Example Question #1 : Use Riemann Sums
Approximate
Using a right Riemann sum, if has the values:
A Riemann sum integral approximation over an interval with
subintervals follows the form:
It is essentially a sum of rectangles, each with a base of length:
and variable heights:
, which depend on the function value at
.
In our case, we have function values over an interval:
Although the total interval has a length of notice the points are equidistant, with
intervals between. Each equidistant interval has length: 5
Since we are using the right point of each interval, we disregard the first function value:
Example Question #7 : Integrating
Approximate
Using a right Riemann sum, if has the values:
A Riemann sum integral approximation over an interval with
subintervals follows the form:
It is essentially a sum of rectangles, each with a base of length:
and variable heights:
, which depend on the function value at
.
In our case, we have function values over an interval:
Although the total interval has a length of notice the points are equidistant, with
intervals between. Each equidistant interval has length:
. Since we are using the right point of each interval, we disregard the first function value:
Example Question #8 : Integrating
The function has the following values on the interval
:
Approximate the integral of on this interval using right Riemann sums.
A Riemann sum integral approximation over an interval with
subintervals follows the form:
It is essentially a sum of rectangles, each with a base of length:
and variable heights:
, which depend on the function value at
.
In our case, we have function values over an interval:
Although the total interval has a length of .
Notice the points are equidistant, with intervals between them. Each equidistance interval has length:
. Since we are using the right point of each interval, we disregard the first function value:
Example Question #9 : Integrating
Approximate
Using a right Riemann sum, if has the values:
A Riemann sum integral approximation over an interval with
subintervals follows the form:
It is essentially a sum of rectangles, each with a base of length:
and variable heights:
, which depend on the function value at
.
In our case, we have function values over an interval:
Although the total interval has a length of notice the points are equidistant, with
intervals between. Each equidistant interval has length:
. Since we are using the right point of each interval, we disregard the first function value:
Example Question #1 : Understand The Fundamental Theorem Of Calculus
Use the fundamental theorem of Calculus to evaluate the definite integral
Here we use the fundamental theorem of Calculus:
Here we do not worry about adding a constant c because we are evaluating a definite integral.
Example Question #1 : Understand The Fundamental Theorem Of Calculus
Evaluate .
We can integrate this without too much trouble
Start
Rewrite the power
Integrate
Evaluate
Simplify
Note that we were not asked to evaluate , so you should not attempt to use part one of the Fundamental Theorem of Calculus. This would give us the incorrect answer of
.
Example Question #481 : Calculus Ab
Using the Fundamental Theorem of Calculus and simplify completely solve the integral.
To solve the integral, we first have to know that the fundamental theorem of calculus is
.
Since denotes the anti-derivative, we have to evaluate the anti-derivative at the two limits of integration, 3 and 6.
To find the anti-derivative, we have to know that in the integral, is the same as
.
The anti-derivative of the function is
, so we must evaluate
.
According to rules of logarithms, when subtracting two logs is the same as taking the log of a fraction of those two values:
.
Then, we can simplify to a final answer of
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