Step 1: Understanding the given region
The region is defined by the following conditions:
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The constraints define a quadrilateral region bounded by the lines:
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- The vertical line
Step 2: Identify the points of intersection
We first find the points where these lines intersect to determine the vertices of the region:
1. Intersection of and :
- Substitute into :
- So, the point of intersection is .
2. Intersection of and :
- Substitute into :
- Now substitute into :
- So, the point of intersection is .
3. Intersection of and :
- Substitute into :
- So, the point of intersection is .
4. Intersection of the vertical line and :
- This is straightforward: .
Step 3: Break the region into smaller parts and calculate the area
The region can be divided into smaller triangles, each having a base and height:
Triangle 1 (bounded by , , and ):
- Base of this triangle is (from to ).
- Height is (from to ).
The area of this triangle is:
Triangle 2 (bounded by , , and ):
- Base of this triangle is .
- Height is (from to ).
The area of this triangle is:
Triangle 3 (bounded by , , and ):
- Base of this triangle is .
- Height is (from to ).
The area of this triangle is:
Step 4: Calculate the total area
The total area of the required region is the sum of the areas of the three triangles: