Find the area of the region bounded by the graphs of the algebraic functions:
f(y)=y^2+12
g(y)=0
y=-12
y=13
To find the area of the region bounded by the graphs of the functions, we need to find the area between two curves.
First, let's sketch a graph of the functions f(y) = y^2 + 12 and g(y) = 0.
The graph of f(y) = y^2 + 12 is a parabola that opens upward. The vertex is at (0, 12), and the graph extends infinitely in both directions.
The graph of g(y) = 0 is simply the x-axis. Since it is a horizontal line at y = 0, it does not intersect the graph of f(y) = y^2 + 12.
Next, let's look at the given vertical lines y = -12 and y = 13. These two lines define the boundaries of the region we want to calculate the area of.
To find the area between the curves, we need to set up an integral. Recall that the area between two curves can be found by integrating the difference of the top curve and the bottom curve with respect to the variable of integration.
Since there is no intersection between the curves f(y) = y^2 + 12 and g(y) = 0, the region corresponds to the area between the curves and the y-axis.
To calculate the area, we integrate the function f(y) = y^2 + 12 over the interval [-12, 13] with respect to y:
Integral[-12 to 13] (y^2 + 12) dy.
Evaluating this integral will give us the area of the region bounded by the given functions.