91Ó°ÊÓ

The region of integration is essentially the part of the xy-plane over which we are evaluating the double integral. In this exercise, the given bounds for the integral are

• \( y \) ranges from 0 to 2
• \( x \) ranges from 0 to \( 4-y^2 \)

These boundaries form a specific shape on the graph. The region can be imagined as the filled area under the curve \( x = 4-y^2 \) with \( y \) spanning from 0 to 2. We can interpret this as part of a parabola opening towards the left on the xy-plane, limited to the first quadrant. This makes the area visually easier to recognize. Drawing a sketch helps to understand how the boundaries enclose the area, making it simpler to comprehend how limits define this space.

Reversing the order of integration involves swapping the roles of \( x \) and \( y \) in the double integral. By doing this, you evaluate the integral in the opposite direction, often simplifying the integration process. To reverse the order:

• Analyze the bounds in terms of \( x \): it originally varies from 0 to \( 4-y^2 \), meaning \( y \) shapes these limits.
• Recharacterize these bounds considering individual values of \( x \) instead.
• Here, \( x \) varies from 0 to 4, and for each \( x \), \( y \) will range from 0 up to \( \sqrt{4-x} \).This transformation involves interpreting the parabola, providing a new perspective for calculating the integral more conveniently.

By redefining limits in terms of the other variable, based on the plotted region of integration, we establish a new integral \( \int_{0}^{4} \int_{0}^{\sqrt{4-x}} y \, dy \, dx \), positioning \( y \), the function being integrated, within the new bounds.

Parabolic bounds occur when the boundary of the region is governed by a quadratic function, commonly forming a parabola on the graph. In this problem,

• The parabola is represented by the equation: \( x = 4-y^2 \).
• This opens to the left with a vertex at \( (4, 0) \).

Visually representing this is crucial: the parabola indicates that for each \( y \), \( x \) can take values starting from 0 to whatever point it touches the parabola. In simpler terms, the parabola dictates how far horizontally the region stretches for each vertical slice (or line) of \( y \). When reversing the integration order, understanding these parabolic limits allows for recalibrating the limits for \( y \) as a function of \( x \), seen in the transformed equation \( y = \sqrt{4-x} \). This insight into how parabolas bound sections of the graph helps to efficiently navigate changing limits and integrate over more intricate shaped regions effectively.