The area of the region bounded by $y = \sin x$ and $y = \cos x$ between $x = \frac { \pi } { 4 }$ and $x = \frac { 5 \pi } { 4 }$ is
A) $2 \sqrt { 2 }$
B) $\frac { \sqrt { 2 } } { 2 }$
C) $\frac { \sqrt { 3 } } { 2 }$
D)0
E) $\sqrt { 2 }$
F) $4 \sqrt { 2 }$
G) $2 \sqrt { 3 }$
H) $\sqrt { 3 }$

Question

The area of the region bounded by $y = \sin x$ and $y = \cos x$ between $x = \frac { \pi } { 4 }$ and $x = \frac { 5 \pi } { 4 }$ is
A) $2 \sqrt { 2 }$ 
B) $\frac { \sqrt { 2 } } { 2 }$ 
C) $\frac { \sqrt { 3 } } { 2 }$ 
D)0
E) $\sqrt { 2 }$ 
F) $4 \sqrt { 2 }$ 
G) $2 \sqrt { 3 }$ 
H) $\sqrt { 3 }$

Quizplus · Accepted Answer

To find the area between $y = \sin x$ and $y = \cos x$ from $x = \frac{\pi}{4}$ to $x = \frac{5\pi}{4}$, we integrate the absolute difference of the two functions over the given interval. The functions intersect at $x = \frac{\pi}{4}$, where they are equal. Between $\frac{\pi}{4}$ and $\frac{5\pi}{4}$, $\cos x$ is less than $\sin x$. Thus, the area is given by the integral from $\frac{\pi}{4}$ to $\frac{5\pi}{4}$ of $(\sin x - \cos x)dx$. This integral evaluates to $2\sqrt{2}$, making choice A correct.

The Area of the Region Bounded By y=sin⁡xy = \sin xy=sinx

The Area of the Region Bounded By $y = \sin x$