Questions: What is the area between the curves (y=cos x+3) and (y=sin x-1) from (x=-pi) to (x=3 pi)? (16 pi) (cos 4 pi) (16 pi+2) (8 pi)

Transcript text: What is the area between the curves $y=\cos x+3$ and $y=\sin x-1$ from $x=-\pi$ to $x=3 \pi ?$ $16 \pi$ $\cos 4 \pi$ $16 \pi+2$ $8 \pi$

Solution

Solution Steps

To find the area between the curves $ y = \cos x + 3 $ and $ y = \sin x - 1 $ from $ x = -\pi $ to $ x = 3\pi $, we need to calculate the definite integral of the absolute difference between the two functions over the given interval. This involves finding the points of intersection within the interval, if any, and integrating the difference of the functions over each subinterval.

Step 1: Define the Functions

We start by defining the two functions given in the problem:

$ f_1(x) = \cos x + 3 $
$ f_2(x) = \sin x - 1 $

Step 2: Find the Points of Intersection

To find the area between the curves, we need to determine the points where the two functions intersect. This occurs when: \[ \cos x + 3 = \sin x - 1 \] Rearranging gives: \[ \cos x - \sin x + 4 = 0 \] We can solve this equation to find the intersection points within the interval $ x = -\pi $ to $ x = 3\pi $.

Step 3: Calculate the Area

The area $ A $ between the curves from $ x = -\pi $ to $ x = 3\pi $ is given by the integral of the absolute difference of the two functions: \[ A = \int_{-\pi}^{3\pi} |f_1(x) - f_2(x)| \, dx \] After evaluating this integral, we find that the area is approximately $ 16\pi $.