The area of the region enclosed by the graphs of the equations F: x = y^2 + 9 and G: 5y = 15 - x, by partitioning the y-axis, is given by the integral of the difference between the upper and lower curves with respect to y. The result, A, represents the area of the region enclosed.
To find the area, we need to determine the y-values at which the curves intersect. By setting the equations F and G equal to each other, we can solve for the values of y. Substituting y^2 + 9 for x in the equation 5y = 15 - x, we have:
5y = 15 - (y^2 + 9)
5y = 15 - y^2 - 9
0 = y^2 + 5y - 6
Factoring the quadratic equation, we have:
0 = (y - 1)(y + 6)
Setting each factor equal to zero, we find two possible y-values: y = 1 and y = -6.
To calculate the area, we integrate the difference between the upper curve (F) and the lower curve (G) with respect to y over the interval [y = -6, y = 1]:
A = ∫[y = -6 to 1] (x upper - x lower) dy
Using the equations F and G, we substitute x = y^2 + 9 for the upper curve and x = 15 - 5y for the lower curve:
A = ∫[y = -6 to 1] [(y^2 + 9) - (15 - 5y)] dy
Evaluating the integral, we find the area A enclosed by the graphs of the equations F and G by partitioning the y-axis.