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To solve the double integral
over the region
bounded by
,
,
, and
, follow these steps:
-
Find Intersection Points:
-
and do not intersect. -
and intersect at . -
and intersect at . -
and intersect at . -
and intersect at . -
and intersect at .
-
-
Determine Region
: - Vertices:
, , .
- Vertices:
-
Set Up the Integral:
- Use the transformation
and . - Jacobian
. - Limits for
from 1 to 4. - Limits for
based on the lines.
- Use the transformation
-
Compute the Integral:
- Determine
and based on the region.
- Determine
-
Evaluate the Integral:
- Integrate with respect to
first. - Then integrate with respect to
.
- Integrate with respect to
After performing the calculations, the value of the integral is
.
Answer: The value of the integral is
.
Solución

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Bonus Knowledge
To find the double integral
over the specified region, we first need to determine the vertices of the region defined by the lines
,
,
, and
.
- Finding Intersection Points:
-
Intersection of
and : Substituting (1) into (2), we have:Point: -
Intersection of
and : Substituting (1) into (2):Point: -
Intersection of
and : Substituting (1) into (2):Point: -
Intersection of
and : Substituting:Point:
-
Final vertices for the region
:
,
,
, and intersects at
.
-
Setting Up the Integral:
The limits of the double integral will be determined by the bounding lines. A transformation can be made usingand , adjusting the Jacobian and integrating accordingly. -
Calculating the Integral:
We will integrateover the transformed bounds once defined clearly through the vertices using a suitable approach, such as Fubini’s Theorem.
In the next step, set proper bounds in each integral according to the area bounded in terms of
and
, and evaluate the definite integral.
The complete evaluation will give a numerical result representing the area under the curve defined by
.
The calculated definite integral will yield the solution to the original problem posed!

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