# calculus

posted by .

find the centroid of the plane region bounded by the curves y = cos x, y=sinx, x=0,

• calculus -

Double check my math, because it gets messy...

Break it into two intervals: [0,pi/4] and [pi/4,pi/2]

Int(sin x)[0,pi/4] + Int(cos x)[pi/4,pi/2]
= (-cos x)[0,pi/4] + (sin x)[pi/4,pi/2]
= 2-√2
This is the denominator in the formulas for xbar and ybar: D = 2-√2

Now for the numerators:
xbarn = Int(x sin x)[0,pi/4] + Int(x cos x)[pi/4,pi/2]
Recall that using integration by parts,
Int(x sin x) = -x cos x + sin x)
Int(x cos x) = x sin x + cos x)

If my math is right, xbarn = pi/4 * (2-√2)

So, xbar = xbarn/D = pi/4
Makes sense, since the area is symmetric about the line x = pi/4.

Now for ybar

ybarn = 1/2 Int(sin^2 x)[0,pi/4] + 1/2 Int(cos^2 x)[pi/4,pi/2]

Recall that
sin^2 x = (1 - cos(2x))/2
cos^2 x = (1 + cos(2x))/2

ybarn = 1/4 Int(1 - cos 2x)[0,pi/4] + 1/4 Int(1 + cos 2x)[pi/4,pi/2]

= 1/4 (x - 1/2 sin 2x)[0,pi/4] + 1/4 (x + 1/2 sin 2x)[pi/4,pi/2]

= 1/8 (pi-2)

So, ybar = ybarn/D = (pi-2)/(8*(2-√2))

## Similar Questions

1. ### MATH

Region A that on xy-plane is bounded by two (2) curves and a line. The curves are y=x^3-2x+3 and y=-x^2+3 while the line is x=0. It is located in the first quadrant of xy-plane. Determine the area of region A.
2. ### Calculus

Hi, I need someone to double check my answer because it doesn't seem right to me. Find the centroid of the region bounded by y=0 and y = x^2 -2x. I calculated the area to be 4/3 and centroid (1, 0.4)... is that correct?
3. ### Calculus [rotation of region bounded by curves]

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=3x^2, x=1, y=0 about the x-axis
4. ### calc

Find the centroid of the region bounded by the given curves. y = 2 sin 3x, y = 2 cos 3x, x = 0, x = π/12
5. ### Math

The curves y=sinx and y=cosx intersects twice on the interval (0,2pi). Find the area of the region bounded by the two curves between the points of intersection.
6. ### Math

The curves y=sinx and y=cosx intersects twice on the interval (0,2pi). Find the area of the region bounded by the two curves between the points of intersection.
7. ### Calculus

find area of the region bounded by the curves y=x^2-1 and y=cos(x). give your answer correct to 2 decimal places.
8. ### Calculus

find the area of the region bounded by the curves y=x^2-1 and y =cos(x)
9. ### Calculus

Find the volume of the solid obtained by rotating the region bounded by the curves y=cos(x), y=0, x=0, and x=π/2 about the line y=1.