How to reverse the order of integration?
Understand the Problem
The question is asking about changing the order of integration in a double integral setup. This situation usually arises in calculus when evaluating integrals in a different order to simplify the computation or to meet certain limits.
Answer
To reverse the order of integration, swap the order of the limits and potentially change the expressions for the limits of integration based on the region.
Answer for screen readers
To reverse the order of integration, follow the detailed steps, sketch the region, and determine the new limits based on the region.
Steps to Solve
- Understand the given region of integration
Analyze the given limits of integration to understand the region over which the integration is taking place. Sketch the region if necessary.
- Sketch the region of integration
Draw the region in the xy-plane to better visualize the boundaries and the limits of integration.
- Determine the new limits of integration
Read off the new limits of integration based on the sketched region. This will usually involve swapping the order of the limits and possibly changing the expressions for the limits of integration.
- Write the new integral
Rewrite the double integral with the new limits of integration.
$$ ext{Old integral:} \ \ \int_{a}^{b} \int_{g(x)}^{h(x)} f(x, y) , dy , dx$$
$$ ext{New integral:} \ \ \int_{c}^{d} \int_{p(y)}^{q(y)} f(x, y) , dx , dy$$
To reverse the order of integration, follow the detailed steps, sketch the region, and determine the new limits based on the region.
More Information
Changing the order of integration can often simplify the computation of double integrals.
Tips
A common mistake is not carefully determining the new limits of integration based on the sketched region.
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