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Example Questions
Example Question #1 : Double Integration Over General Regions
Calculate the following Integral.
Lets deal with the inner integral first.
Now we evaluate this expression in the outer integral.
Example Question #2 : Double Integration Over General Regions
Calculate the definite integral of the function , given below as
Cannot be solved.
Because there are no nested terms containing both and
, we can rewrite the integral as
This enables us to evaluate the double integral and the product of two independent single integrals. From the integration rules from single-variable calculus, we should arrive at the result
.
Example Question #1 : Double Integration Over General Regions
Evaluate the following integral on the region specified:
Where R is the region defined by the conditions:
Example Question #1 : Double Integration Over General Regions
Evaluate:
Because the x and y terms in the integrand are independent of one another, we can move them to their respective integrals:
We used the following rules for integration:
,
Example Question #3 : Double Integration Over General Regions
Evaluate the following integral.
First, you must evaluate the integral with respect to y (because of the notation ).
Using the rules of integration, this gets us
.
Evaluated from y=2 to y=3, we get
.
Integrating this with respect to x gets us , and evaluating from x=0 to x=1, you get
.
Example Question #4 : Double Integration Over General Regions
Compute the following integral:
First, you must evaluate the integral with respect to y and solving within the bounds.
In doing so, you get and you evaluate for y from 0 to 2.
This gets you
.
This time evaluating the integral with respect to x gets you
.
Evaluating for x from 1 to 2 gets you
.
Example Question #7 : Double Integration Over General Regions
Evaluate the double integral.
When solving double integrals, we compute the integral on the inside first.
Example Question #1 : Double Integration Over General Regions
Evaluate the double integral.
When solving double integrals, we compute the integral on the inside first.
Example Question #6 : Double Integration Over General Regions
Evaluate the double integral
When solving double integrals, we compute the integral on the inside first.
Example Question #7 : Double Integration Over General Regions
Evaluate the integral
First, you must evaluate the integral with respect to x. This gets you evaluated from
to
. This becomes
. Solving this integral with respect to y gets you
. Evaluating from
to
, you get
.
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