Solve the problem.
-Find the centroid of the region in the first quadrant bounded by the curve y = 7 cos x and the axes.
A) $\bar { x } = \frac { \pi - 2 } { 2 } , \bar { y } = \frac { 7 \pi } { 8 }$
B) $\bar { x } = \frac { 7 \pi } { 8 } , \bar { y } = \frac { \pi - 2 } { 2 }$
C) $\bar { x } = \frac { \pi } { 2 } , \bar { y } = \frac { 7 \pi } { 4 }$
D) $\bar { x } = \frac { 7 \pi - 14 } { 2 } , \bar { y } = \frac { 49 \pi } { 8 }$

Question

Solve the problem.
-Find the centroid of the region in the first quadrant bounded by the curve y = 7 cos x and the axes. 
A) $\bar { x } = \frac { \pi - 2 } { 2 } , \bar { y } = \frac { 7 \pi } { 8 }$
B) $\bar { x } = \frac { 7 \pi } { 8 } , \bar { y } = \frac { \pi - 2 } { 2 }$
C) $\bar { x } = \frac { \pi } { 2 } , \bar { y } = \frac { 7 \pi } { 4 }$
D) $\bar { x } = \frac { 7 \pi - 14 } { 2 } , \bar { y } = \frac { 49 \pi } { 8 }$

Quizplus · Accepted Answer

The formula for the x-coordinate of the centroid is given by: 
$$\bar {x} = \frac{1}{A} \int_{a}^{b} x f(x) dx$$
where A is the area of the region and f(x) is the equation of the curve.

The formula for the y-coordinate of the centroid is given by: 
$$\bar {y} = \frac{1}{A} \int_{a}^{b} \frac{f(x)^2}{2} dx$$
where A is the area of the region and f(x) is the equation of the curve.

Using these formulas, we get: 
$$\bar {x} = \frac{1}{7\pi/2} \int_{0}^{\pi/2} x\cdot 7\cos(x) dx = \frac{\pi-2}{2}$$
$$\bar {y} = \frac{1}{7\pi/2} \int_{0}^{\pi/2} \frac{(7\cos(x))^2}{2} dx = \frac{7\pi}{8}$$

Therefore, the centroid of the region is $\left(\frac{\pi-2}{2},\frac{7\pi}{8}\right)$, which corresponds to choice A.

Solve the Problem xˉ=π−22,yˉ=7π8\bar { x } = \frac { \pi - 2 } { 2 } , \bar { y } = \frac { 7 \pi } { 8 }xˉ=2π−2​,yˉ​=87π​

Solve the Problem $\bar { x } = \frac { \pi - 2 } { 2 } , \bar { y } = \frac { 7 \pi } { 8 }$