Double integral of ln(1+x^2+y^2) over the region x^2+y^2 ≤ e^2-1

Published on December 25, 2024

Calculus

The double integral, using polar coordinates and integration by parts, evaluates to π(e^2 + 1).

Question

∫∫ D ln(1+x^2 +y^2 )dxdy D = x^2 + y^2 ≤ e^2 - 1

Answer

Here’s how to solve the double integral:

1. Change to Polar Coordinates:

Given the region D defined by

, and the integrand involving

, it’s best to switch to polar coordinates. Recall the transformations:

The region D in polar coordinates becomes:

The integral transforms to:

2. Solve the Inner Integral:

Let’s focus on the inner integral:

Use the substitution

, so

. The limits of integration change as well:

When , .
When , .

The inner integral becomes:

Now, integrate by parts: Let

and

. Then

and

Evaluating at the limits:

3. Solve the Outer Integral:

Now we have:

This is a simple integral:

Final Answer:

The value of the double integral is

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