91Ó°ÊÓ

In double integrals, the initial order in which integration is performed matters.The original exercise had the order being integrated as follows: first with respect to \( x \), then to \( y \).To interchange the order means to first perform the integration with respect to \( y \) and then to \( x \).This is crucial because it can affect the integration process and sometimes make it simpler.

When changing the order of integration, the function being integrated and the limits of integration may need to be re-evaluated, ensuring they are correctly re-aligned with the new order.This involves re-examining the region of integration, sometimes requiring a sketch to visualize the area more clearly.
Note that the final result of the integral should ultimately remain unchanged, as the value of the definite integral should be independent of the order of integration.

Identifying the integration limits is essential when solving double integrals.The original limits, \( y \) from 0 to 2 and \( x \) from 0 to 1, were given for the initial setup.

After interchanging the order of integration:

• The limits for \( x \) remained from 0 to 1 as they were already straightforward.
• The limits for \( y \), originally independent of \( x \), stayed the same from 0 to 2.

Preserving these limits accurately is crucial, as they define the rectangular region over which the double integral is evaluated.
These limits help determine the span of the integration in their respective directions, effectively bounding the region over which you are integrating.

Calculating the inner integral is where you break down the integral into parts, making integration manageable.After you've interchanged the order of integration, you deal with the inner integral first.

In this exercise, the inner integral is conducted with respect to \( y \):\[ \int_{0}^{2} (x + 2e^y - 3) \, dy \]To solve:

• Integrate each term separately. This decomposes the integral into bite-sized pieces.
• For \( x \), treat it as a constant: \( x [y]_{0}^{2} \).
• For \( 2e^y \), apply the exponential rule: \( 2[e^y]_{0}^{2} \).
• For the constant \( 3 \), use simplicity: \( 3[y]_{0}^{2} \).

This process results in an expression that becomes the subject of the subsequent integral involving \( x \).
Inner integral calculation often involves ensuring that you accurately alter variables and constants to fit the new order of integration.

Simplifying the integral once the inner integral has been calculated is the next step.This ensures you deal with an easy-to-integrate expression later on.

From the example, after performing the inner integration, you got:\[ 2x + 2(e^2 - 1) - 6 \]This can further be simplified to:\[ 2x + 2e^2 - 8 \]
This simplification process involves clearing out brackets, combining like terms, and making the expression as succinct as possible before proceeding to the second integration.
It leads to a much more efficient integration process for the final integral, making sure you can focus on simpler mathematical operations in subsequent steps.