Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region.
y=e^(4x), y=e^(6x), x=1
To sketch the region enclosed by the given curves, we need to find the intersection points of the curves.
First, let's find the intersection points between y = e^(4x) and y = e^(6x):
e^(4x) = e^(6x)
Take the natural logarithm of both sides to eliminate the exponentials:
4x = 6x
2x = 0
x = 0
So, the curves intersect at x = 0.
Next, let's determine whether to integrate with respect to x or y. To do this, we need to visualize the region.
Let's graph the two curves on the same coordinate plane:
By observing the graph, we see that the curve y = e^(6x) is always above y = e^(4x) for x ≥ 0. Therefore, the region is bounded by the curves y = e^(4x) and y = e^(6x), and the lines x = 0 and x = 1.
To find the area of this region, we will integrate with respect to x.
The integral that represents the area of the region is:
A = ∫[0,1] (e^(6x) - e^(4x)) dx
To calculate this integral, we can use the properties of exponential functions and the power rule of integration.
A = [1/6 * e^(6x) - 1/4 * e^(4x)] evaluated from 0 to 1
A = [1/6 * e^(6 * 1) - 1/4 * e^(4 * 1)] - [1/6 * e^(6 * 0) - 1/4 * e^(4 * 0)]
A = (1/6 * e^6 - 1/4 * e^4) - (1/6 * 1 - 1/4 * 1)
A = (1/6 * e^6 - 1/4 * e^4) - (1/6 - 1/4)
Finally, calculate the numerical value of A to find the area of the region.