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By changing the order of integration in 
the value is
the value is- a)π/4
- b)πa/4
- c)πa
- d)a/4
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
By changing the order of integration inthe value isa)π/4b)πa/4c)...
To evaluate the integral:
∫0a ∫ya (x / (x² + y²)) dx dy
by changing the order of integration, we need to determine the region of integration in the x-y plane and then rewrite the integral accordingly.
Step 1: Determine the Region of Integration
The given limits indicate that:
- y ranges from 0 to a,
- For a fixed y, x ranges from y to a.
In the x-y plane, this describes the triangular region bounded by:
- y = 0,
- x = a,
- x = y.
Step 2: Change the Order of Integration
To change the order of integration, we need to describe the region in terms of x first:
- x ranges from 0 to a,
- For a fixed x, y ranges from 0 to x (since y ≤ x within this region).
Thus, we can rewrite the integral as:
Step 3: Evaluate the Inner Integral with Respect to y
Now, we evaluate the inner integral:
Since x is treated as a constant in the inner integral, we can factor it out:
Integrate with respect to y:
Since arctan(1) = π/4 and arctan(0) = 0, this becomes:
= (1/x) * (π/4) = π / (4x).
Step 4: Substitute Back and Integrate with Respect to x
Now our integral becomes:
This simplifies to:
Conclusion:
The value of the integral is:
B: πa / 4.
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