Final answer:
To find the area of the region enclosed by the graphs of the functions f(x)=x and g(x)=x¹⁵, we can partition the x-axis and integrate with respect to x. The area of the region is found by calculating the integral of the difference between the two functions over the specified interval. In this case, the area is -7/16.
Step-by-step explanation:
To find the area of the region enclosed by the graphs of the functions f(x)=x and g(x)=x¹⁵, we can partition the x-axis and integrate with respect to x. The region enclosed by the two graphs will be the area between the curves.
Let's find the intersection points of the two graphs by setting f(x) equal to g(x):
x = x¹⁵
Simplifying the equation, we get:
x - x¹⁵ = 0
Factoring out an x, we have:
x(1 - x¹⁴) = 0
The graph intersects at x = 0 and x = 1. Now, we can integrate to find the area:
A = ∫[0,1] (g(x) - f(x)) dx
Using the power rule of integration, we have:
A = ∫[0,1] (x¹⁵ - x) dx
Integrating, we get:
A = [x¹⁶/16 - x²/2] evaluated from 0 to 1
Plugging in the limits of integration:
A = [(1/16) - (1/2)] - [(0/16) - (0/2)]
A = 1/16 - 1/2
A = -7/16
Therefore, the area of the region enclosed by the graphs of the functions f(x)=x and g(x)=x¹⁵ is -7/16.