Question
QUIZ 2 Sample
ANSWER ALL QUESTIONS
Q1. FSCE 1993 (ENG MATH 2)
and
. ANS 128 of
if
is the region bounded by lines
ANSWER ALL QUESTIONS
Q1. FSCE 1993 (ENG MATH 2)
and
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:
- For
: gives the point . - 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:
- Evaluate
:
- 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
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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:
- The lines
and create a V-shape, while the line caps the area above these lines. - 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:
- For
:
- For
:
Now compile the inner integral:
Next, we integrate this with respect to
:
Calculating each integral:
- For
:
- For
:
Plugging these back into our equation:
Converting
to fifths:
Finally:
Thus, the final answer is: