Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y, y=4sin(x), y=e^(7x), x=0, x=pi/2
For all x: 0<=x<=pi/2
4sin(x)<e^(7x)
We begin to integrate with respect to y:
Int.[from 4sin(x) to e^(7x)]...dy
To sketch the region enclosed by the given curves, we need to find the points of intersection between the curves and determine the boundaries of the region.
The given curves are:
1. y = 4sin(x)
2. y = e^(7x)
To find the points of intersection, set the two equations equal to each other:
4sin(x) = e^(7x)
To solve this equation, we can use a numerical method such as graphing both functions and finding the points of intersection or use software like Wolfram Alpha.
Once you find the points of intersection, let's denote them as A and B. We'll also consider the vertical lines x = 0 and x = π/2 as the boundaries of the region.
To determine whether to integrate with respect to x or y, we need to see which variable bounds the region vertically. Looking at the given curves, we can see that the curve y = 4sin(x) is above y = e^(7x) within the interval [0, π/2]. This means that the region is bounded above and below by the curves, so we should integrate with respect to x.
The region can be defined as the area between the curves y = 4sin(x) and y = e^(7x) from x = 0 to x = π/2.
To find the area of the region, we need to evaluate the integral of the difference between the two curves over the given interval:
Area = ∫[a, b] (top curve - bottom curve) dx
Using the boundaries of x = 0 and x = π/2, the integral becomes:
Area = ∫[0, π/2] (e^(7x) - 4sin(x)) dx