91Ó°ÊÓ

In double integrals, the region of integration is crucial as it defines the area over which we calculate the integral. For the given problem, the region is defined by the integration limits of both \(x\) and \(y\). The original inner integral, \(\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\), specifies that for each fixed value of \(y\), \(x\) varies between \(-\sqrt{1-y^2}\) and \(\sqrt{1-y^2}\).

This range describes a horizontal spread across a semicircle. Since \(y\) ranges from 0 to 1, the semicircle is the upper half, centered at the origin, with a radius of 1.

• When \(y = 0\), \(x\) ranges from \(-1\) to 1.
• When \(y = 1\), \(x = 0\).

Each point within this region gets accounted for in the integration process, ensuring the integral accurately reflects the area or volume it intends to measure or calculate.

Reversing the order of integration is a common technique in calculus that helps simplify computation or make the integral solvable. To reverse the order of integration, we need to describe the same region, but with \(y\) being integrated first and \(x\) second.

To achieve this, observe the same semicircle region, but consider \(x\) to define the outer limits. Here, \(x\) ranges from \(-1\) to 1, encompassing the base of our semicircle.

• For a given \(x\), \(y\) will vary from the bottom edge (\(y = 0\)) up to the arc of the semicircle (\(y = \sqrt{1-x^2}\)).

By correctly identifying how \(y\) extends for any fixed \(x\), we ensure we describe the entire semicircular area. This thoughtful adjustment allows us to interchange the order of integration without altering the region's boundary.

Integration limits play a central role in defining and evaluating double integrals. They specify the bounds within which we integrate a function. Initially, our limits \(\int_{0}^{1} \, dy\) and \(\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} \, dx\) define a specific region. This is a key aspect because these values establish the shape and size of the area we are dealing with.

Once we decide to reverse the order of integration, the limits change to \(\int_{-1}^{1} \, dx\) and \(\int_{0}^{\sqrt{1-x^2}} \, dy\).

• For the reversed order, \(x\) varies across the diameter of the semicircle, \(-1\) to \(1\).
• For any fixed value of \(x\) in this range, \(y\) smoothly extends from 0 up to the boundary \(\sqrt{1-x^2}\).

Getting these limits right ensures that the integral covers the full region intended, without missing any area or counting any extra. Hence, modifying limits when reversing integration order maintains the integrity of the integral's original purpose.