To change the order of integration for the given double integral:

$\int_{0}^{2\pi} \int_{-1}^{\cos x} f(x, y) \, dy \, dx,$

we need to analyze the region of integration and determine how to reverse the order of integration.

Step 1: Understand the region of integration

The bounds for $y$ and $x$ are given as:

• $y$ varies from $-1$ to $\cos x$.
• $x$ varies from $0$ to $2\pi$.

To rewrite this, the region must be expressed in terms of $x$ as a function of $y$.

Step 2: Rewrite the region

The condition $y \leq \cos x$ implies that: $x = \cos^{-1}(y),$ where $\cos^{-1}$ is the inverse cosine function, valid for $y \in [-1, 1]$.

The region of integration is:

1. $y$ ranges from $-1$ to $1$, based on the range of the cosine function.
2. For each fixed $y$, $x$ varies from $\cos^{-1}(y)$ to $2\pi - \cos^{-1}(y)$, considering symmetry in the cosine function.

Step 3: Change the order of integration

The integral in terms of the new bounds is: $\int_{-1}^{1} \int_{\cos^{-1}(y)}^{2\pi - \cos^{-1}(y)} f(x, y) \, dx \, dy.$

Would you like me to visualize the integration region or provide additional clarification? Let me know!

5 Related Questions

1. How is the region of integration determined for other trigonometric functions?
2. What is the relationship between the bounds of integration and inverse functions?
3. How can this method be generalized for polar coordinates?
4. What are common errors when reversing the order of integration?
5. How does symmetry simplify the process of changing integration order?

Tip

When reversing integration order, always sketch the region and understand its geometric structure before attempting to rewrite bounds.