![## Evaluating a Double Integral
### Problem Statement
**2) Evaluate the integral:**
\[
\iint_{-2}^{2} \int_{0}^{1} \frac{xy}{1 + y^4} \, dy \, dx
\]
### Explanation
This double integral involves evaluating an iterated integral over a specified region. The region of integration is defined by \(x\) ranging from \(-2\) to \(2\), and for each fixed \(x\), \(y\) ranging from \(0\) to \(1\). The integrand is the function \(\frac{xy}{1 + y^4}\).
#### Steps for Evaluation:
1. **Inner Integral (with respect to \(y\)):**
- Integrate \(\frac{xy}{1 + y^4}\) with respect to \(y\) from \(0\) to \(1\).
2. **Outer Integral (with respect to \(x\)):**
- After evaluating the inner integral, integrate the resulting expression with respect to \(x\) from \(-2\) to \(2\).
### Additional Notes
- This problem requires knowledge of techniques for evaluating definite integrals, particularly when involving products and compositions of functions.
- Checking for potential symmetries or simplifications might be useful before performing tedious calculations.
- The final result will provide the accumulated "area" under the surface defined by \(\frac{xy}{1 + y^4}\) over the specified region.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F10580cf2-5676-4d37-9162-c9dfcc15f202%2F9bb55f85-e9c6-4fcf-9de7-0730a491b865%2Fylogllr_processed.jpeg&w=3840&q=75)
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I now know about the even function rule where you can change the bounds from -a to a to 0 to a by putting a 2 out front of the integral for any "even function." After asking my first question about that the new solution did not use that method and instead used -2 to 2 as in the original problem, this makes the answer 0 and if you use the even function method and pull the 2 out front the answer is pi/2. Which of these is the correct answer? How do we know when we can use that even function rule or when not to. I noticed the rule said an even function is when f(-x)=f(x) but I'm not exactly sure what that means
by Bartleby ExpertWhy can we pull the 2 out front and change the bounds of the
by Bartleby Expert
