Let R be the first quadrant region enclosed by y=e^2, y=e^x and the y-axis. The area of region R is?
I don't understand why is the interval is from [0-2].
Thanks
You need the intersection of y = e^x and y = e^2
e^x = e^2
so x = 2
then area = ∫ e^2 - e^x from 0 to 2
= [e^2 x - e^x] from 0 to 2
= 2e^2 - e^2 - (0 - e^0)
= e^2 + 1
To find the area of region R, we need to determine the limits of integration for the integral that will be used to calculate the area.
The given functions, y = e^2, y = e^x, and the y-axis, form the boundaries of region R in the first quadrant.
The interval [0, 2] is used because we are looking for the values of x that determine the bounds of region R. Since y = e^2 is constant, it intersects the y-axis at y = e^2 when x is 0. And since y = e^x is an increasing function, it intersects the y-axis at y = 1 (e^x = 1, x = 0), and it intersects y = e^2 when x = 2 (e^x = e^2).
Therefore, the interval [0, 2] encompasses the x-values where the boundaries of region R intersect the y-axis. This interval represents the range of x-values over which we need to integrate to find the area of region R.