91Ó°ÊÓ

The inverse sine function, often written as \( \sin^{-1}y \) or arcsin, is an essential concept in trigonometry. It helps you find the angle whose sine is a given number. For example, if \( \sin^{-1}(0.5) = \theta \), then it means \( \sin(\theta) = 0.5 \). The values of \( y \) for the inverse sine function range from \(-1\) to \(1\). This is because the sine of an angle cannot exceed these values. Hence, \( \sin^{-1}(y) \) is defined for \( y \) between \(-1\) and \(1\).
In our given exercise, the limit \( y \) is between \(0\) and \(1\), implying \( x \) is bounded between \(0\) and \( \pi/2 \) since \( \sin^{-1}(0) = 0 \) and \( \sin^{-1}(1) = \pi/2 \). The inverse sine function is crucial in finding this sector of a circle in the coordinate plane.

The coordinate plane is a two-dimensional surface where you can represent mathematical figures. It consists of two lines known as axes: the horizontal line called the x-axis, and the vertical line called the y-axis. When you define a region for integration as in this exercise, you're dealing with a specific portion of this plane.
Each point on a coordinate plane is represented as \((x, y)\), where \(x\) and \(y\) are the horizontal and vertical distances from the origin \((0, 0)\). For our task, the integration limits create a section of the plane that resembles a wedge.
When you sketch a region like in this particular exercise, you start with the range for \( y \) between \(0\) and \(1\) on the y-axis. At each instance of \( y \), \( x \) spans from \(0\) to \( \sin^{-1}y \), gradually expanding along the x-axis as \( y \) increases. This results in a growing boundary forming enclosed by the x-axis, y-axis, and the curve defined by \( x = \sin^{-1} y \).

Graphical representation provides a visual understanding of functions and their behavior in a defined region. By plotting the function \( x = \sin^{-1}y \), you visualize how the function behaves from \( y = 0 \) to \( y = 1 \). Begin with critical points:\

• At \( y = 0 \), \( x \) also starts at 0.
• At \( y = 1 \), \( x \) reaches \( \pi/2 \).

These points are verified by the evaluation of \( \sin^{-1}(y) \) at its bounding values. The curve accented by these evaluations is essential in defining the region over which integration will occur.
To sketch, plot the linear boundaries along the x and y axes, and then carefully draw the curved line of \( x = \sin^{-1} y \) connecting \((0, 0)\) steadily rising to \((\pi/2, 1)\). The area is below this curve, above the x-axis, providing a visible illustration of the function’s range. This sketch aids in grasping the spatial characteristics of the mathematical equation and thoroughly understanding the area you have restricted for integration.