What is the area of the region between $2$ consecutive polnts where the graphs of $f(x)=\cos (x)$ and $g(x)=-\cos (x)+2$ intersect? $\newline$ Choose $1$ answer: $\newline$ (A) $4$ $\newline$ (B) $4 \pi$ $\newline$ (C) $2$ $\newline$ (D) $2 \pi$

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Solve equations with variables on both sides: word problems

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Q. What is the area of the region between

2

consecutive polnts where the graphs of

f(x)=\cos (x)

and

g(x)=-\cos (x)+2

intersect?

\newline

Choose

1

answer:

\newline

(A)

4

\newline

(B)

4 \pi

\newline

(C)

2

\newline

(D)

2 \pi

Find Intersection Points: First, we need to find the points of intersection between the two functions $f(x) = \cos(x)$ and $g(x) = -\cos(x) + 2$ .
Set Equations Equal: To find the points of intersection, we set $f(x)$ equal to $g(x)$ : $\cos(x) = -\cos(x) + 2$
Solve for x: Solving for x, we add $\cos(x)$ to both sides of the equation: $\newline$ $\cos(x) + \cos(x) = 2$ $\newline$ $2\cos(x) = 2$
Identify $x$ Values: Divide both sides by $2$ to isolate $\cos(x)$ : $\cos(x) = 1$
Calculate Areas: The value of $x$ for which $\cos(x) = 1$ is $x = 2n\pi$ , where $n$ is an integer. However, we are looking for the first two consecutive points of intersection, so we will consider $n = 0$ and $n = 1$ .
Determine Total Area: The first point of intersection is at $x = 0$ , and the second point of intersection is at $x = 2\pi$ . These are the consecutive points where the two graphs intersect.
Determine Total Area: The first point of intersection is at $x = 0$ , and the second point of intersection is at $x = 2\pi$ . These are the consecutive points where the two graphs intersect.Now, we need to find the area of the region between these two points. The graphs of $f(x)$ and $g(x)$ are symmetrical about the line $y = 1$ , and the area between them from $x = 0$ to $x = 2\pi$ is a pair of identical ＂bumps＂ above and below $y = 1$ .
Determine Total Area: The first point of intersection is at $x = 0$ , and the second point of intersection is at $x = 2\pi$ . These are the consecutive points where the two graphs intersect.Now, we need to find the area of the region between these two points. The graphs of $f(x)$ and $g(x)$ are symmetrical about the line $y = 1$ , and the area between them from $x = 0$ to $x = 2\pi$ is a pair of identical ＂bumps＂ above and below $y = 1$ .The area of one ＂bump＂ above $y = 1$ is the same as the area of one ＂bump＂ below $y = 1$ . Since the total area from $x = 0$ to $x = 2\pi$ under the curve $x = 2\pi$ $2$ is $x = 2\pi$ $3$ (the integral of $x = 2\pi$ $4$ from $0$ to $x = 2\pi$ $3$ ), the area of one ＂bump＂ is half of that, which is $x = 2\pi$ $6$ .
Determine Total Area: The first point of intersection is at $x = 0$ , and the second point of intersection is at $x = 2\pi$ . These are the consecutive points where the two graphs intersect.Now, we need to find the area of the region between these two points. The graphs of $f(x)$ and $g(x)$ are symmetrical about the line $y = 1$ , and the area between them from $x = 0$ to $x = 2\pi$ is a pair of identical ＂bumps＂ above and below $y = 1$ .The area of one ＂bump＂ above $y = 1$ is the same as the area of one ＂bump＂ below $y = 1$ . Since the total area from $x = 0$ to $x = 2\pi$ under the curve $x = 2\pi$ $2$ is $x = 2\pi$ $3$ (the integral of $x = 2\pi$ $4$ from $0$ to $x = 2\pi$ $3$ ), the area of one ＂bump＂ is half of that, which is $x = 2\pi$ $6$ .Therefore, the total area of the region between the two graphs from $x = 0$ to $x = 2\pi$ is twice the area of one ＂bump,＂ which is $x = 2\pi$ $9$ .

What is the area of the region between 222 consecutive polnts where the graphs of f(x)=cos⁡(x) f(x)=\cos (x) f(x)=cos(x) and g(x)=−cos⁡(x)+2 g(x)=-\cos (x)+2 g(x)=−cos(x)+2 intersect?\newlineChoose 111 answer:\newline(A) 444\newline(B) 4π 4 \pi 4π\newline(C) 222\newline(D) 2π 2 \pi 2π

What is the area of the region between $2$ consecutive polnts where the graphs of $f(x)=\cos (x)$ and $g(x)=-\cos (x)+2$ intersect? $\newline$ Choose $1$ answer: $\newline$ (A) $4$ $\newline$ (B) $4 \pi$ $\newline$ (C) $2$ $\newline$ (D) $2 \pi$