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How to solve the b) question?? Mind to guide me?? and what you get for the final answer ?? Thanks ^^
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Integrating with respect to z gives 4xyz.
Evaluating from x downto 0 gives 4x²y - 0 = 4x²y.
Integrating with respect to y next gives 2x²y²,
Evaluating from x downto √x gives 2x^4 - 2x³.
Integrating with respect to x gives (2x^5)/5.
Evaluating this from 1 downto 0 gives 2/5.
The next problem is known can be converted to polar coordinates.
Let r² = x² + y² and A be the angle. To do this, note that ∫∫ dx dy = ∫∫ r dr dA,
where r is the radius and A is the angle. This gives ∫∫ r / (r²+1)² dr dA.
Note that r goes from 3 downto 1 and A goes from π/2 downto π/4.
Letting s = r+1, ds = dr and r = s - 1. This gives ∫∫(s-1)/s² ds dA.
That can be rewritten as ∫∫1/s - 1/s² ds dA.
The integral with respect to s of this is ln(s) + 1/s.
Since r went from 3 downto 0, and s = r+1, s goes from 4 down to 1.
This gives ln(4) + 1/4 - ln(1) - 1 = ln(4) - 3/4 = ln(4) - 0.75.
When we integrate with respect to A, we get (ln(4) - 0.75)A.
Since A went from π/2 down to π/4, the difference is π/4.
This makes the answer be π(ln(4) - 0.75)/4.
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