To sketch the region bounded by the graphs of the given functions and find its area, let's follow the following steps:
Step 1: Determine the boundaries of the region.
The given functions are f(y) = y/â(16-y^2), g(y) = 0, and y = 3. We need to find the range of y-values for which the region is defined.
Since y = 3 is given as one of the boundaries, we know that the region extends up to this line. Additionally, the graphs of f(y) and g(y) define the upper and lower boundaries of the region, respectively.
To find the range of y-values for which the region exists, we can analyze the first function, f(y), since g(y) = 0 is a constant. The denominator of f(y) should not be zero, so 16 - y^2 â 0. Solving this equation, we find y â Âą4.
Therefore, the region is defined for -4 < y < 4.
Step 2: Sketch the graphs of f(y) and g(y).
To sketch the graph of f(y), we need to determine its behavior as y changes. Notice that f(y) involves a square root function, which is always nonnegative. We can start by analyzing the behavior of f(y) as y approaches the boundary values.
1. If y approaches -4 from the left (y â â4-), then 16 - y^2 will be slightly greater than zero. Since the denominator contains the square root of this value, f(y) will be finite and positive.
2. As y approaches 4 from the right (y â 4+), 16 - y^2 becomes slightly greater than zero. Hence, f(y) will be finite and positive as well.
Based on these observations, we can conclude that the graph of f(y) starts from a positive value for y = -4, increases as y approaches 0, reaches a maximum, and then decreases as y approaches 4. It is symmetric about the y-axis.
On the other hand, the graph of g(y) is just a horizontal line at y = 0.
Step 3: Shade the region bounded by the graphs of f(y) and g(y).
Based on the descriptions above, the region bounded by the graphs of f(y) and g(y) can be visualized as a closed interval on the y-axis between y = -4 and y = 4, enclosed by the graph of f(y) and the horizontal line y = 0.
Step 4: Calculate the area of the region.
To find the area of the shaded region, we need to integrate f(y) with respect to y over the interval for which the region is defined (from -4 to 4):
Area = âŤ[from -4 to 4] f(y) dy.
However, since f(y) is a function that involves a square root, integrating it directly may not be straightforward. Instead, we can use symmetry to simplify the integration.
Since f(y) is symmetric about the y-axis, the positive and negative portions of the graph have equal areas. So, we can find the area of the right half of the region and then double it to obtain the total area.
Thus, the area of the region is:
Area = 2 * âŤ[from 0 to 4] f(y) dy.
Evaluating this integral will give you the exact area of the region.
Note: Calculating the integral may require advanced techniques or numerical methods, depending on the difficulty of integrating the function f(y).