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QUIZ 2 Sample
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Q1. FSCE 1993 (ENG MATH 2)
and . ANS 128 of if is the region bounded by lines

Ask by King Erickson. in Malaysia
Dec 07,2024

Upstudy AI Solution

Tutor-Verified Answer

Answer

The value of the double integral is 128.

Solution

To evaluate the double integral
where is the region bounded by the lines and and the line , we first need to determine the region .

Step 1: Identify the region

The lines and intersect the line at the points:
  1. For : gives the point .
  2. For : gives the point .
Thus, the region is bounded by the lines , , and the horizontal line . The vertices of the region are , , .

Step 2: Set up the double integral

The region can be described in terms of from to and for each fixed , ranges from to . Therefore, we can set up the double integral as follows:

Step 3: Evaluate the inner integral

We first evaluate the inner integral:
  1. Evaluate :
  1. Evaluate :
Combining these results, we have:

Step 4: Evaluate the outer integral

Now we evaluate the outer integral:

Final Answer

Thus, the value of the double integral is

Answered by UpStudy AI and reviewed by a Professional Tutor

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Bonus Knowledge

To solve the double integral over the region defined by the lines , , and the line , we first visualize the region:
  1. The lines and create a V-shape, while the line caps the area above these lines.
  2. The region of integration is thus a triangle with vertices at the points , , and .
Now we can set up the limits for the double integral. We can opt to integrate with respect to first, where goes from to , and goes from to :
Calculating the inner integral:
  1. For :
  1. For :
Now compile the inner integral:
Next, we integrate this with respect to :
Calculating each integral:
  1. For :
  1. For :
Plugging these back into our equation:
Converting to fifths:
Finally:
Thus, the final answer is:

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