Math Problem Statement
answer
Solution
The problem asks to compute the double integral over the region , which is defined as .
Step 1: Describe the Region
The region is the first quadrant portion of a circle centered at the origin with radius (since ).
Step 2: Set Up the Integral in Polar Coordinates
It's convenient to use polar coordinates, where:
The region in polar coordinates has:
The integral becomes:
Step 3: Simplify the Integral
Simplify the integrand:
So the integral becomes:
Step 4: Evaluate the Integral
Separate the integral:
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Evaluate the radial part:
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Evaluate the angular part: Use the identity :
Step 5: Multiply the Results
The final answer is:
However, it seems there might have been a mistake in evaluating the trigonometric integral. The correct factor is actually: So, the correct multiplication should be:
Finally, consider the closest answer to this result from the given options: The actual answer corresponds to .
Would you like a more detailed explanation on any part?
Here are 5 related questions for further exploration:
- How do you convert a double integral in Cartesian coordinates to polar coordinates?
- What are common mistakes when evaluating trigonometric integrals?
- How does symmetry simplify the evaluation of integrals over circular regions?
- What are some practical applications of double integrals in physics or engineering?
- Can double integrals be solved using numerical methods, and when is that necessary?
Tip: When setting up integrals in polar coordinates, always remember to include the Jacobian determinant, which in 2D is .
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Math Problem Analysis
Mathematical Concepts
Double Integrals
Polar Coordinates
Trigonometric Identities
Formulas
x = r * cos(θ), y = r * sin(θ)
dx dy = r dr dθ
∫∫_R f(x, y) dx dy = ∫∫_R f(r cos(θ), r sin(θ)) * r dr dθ
Theorems
Change of Variables in Double Integrals
Suitable Grade Level
Undergraduate (Calculus II/III)
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