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Let Rbe rectangular region with vertices (0,0),(b,0),(0,h),and(b,h)

Find the centroid ofR

Short Answer

Expert verified

The centroid isxยฏ=b2,yยฏ=h2

Step by step solution

01

Given Information

Vertices of rectangular region is(0,0),(b,0),(0,h)and(b,h)

02

Calculation of x-

Centroid is expressed as xยฏ=โˆฌฮฉxฯ(x,y)dAโˆฌฮฉฯ(x,y)dAandyยฏ=โˆฌฮฉyฯ(x,y)dAโˆฌฮฉฯ(x,y)dA

Density is uniform, ฯ(x,y)=1

xยฏ=โˆซ0bโˆซ0hxdydxโˆซ0bโˆซ0hdydx

Solving inner integral first

xยฏ=โˆซ0bx[y]0hdxโˆซ0b[y]0hdx

xยฏ=โˆซ0bxdxโˆซ0bdx

Using limits xยฏ=โˆซ0bxdxโˆซ0bdx

Solving further

xยฏ=x220b[x]0b=b22[b]

xยฏ=b2

03

Calculation of y-

Similarly

yยฏ=โˆฌฮฉyฯ(x,y)dAโˆฌฮฉฯ(x,y)dA=โˆซ0bโˆซ0hydydxโˆซ0bโˆซ0hdydx=โˆซ0by220hdxโˆซ0b[y]0hdx=โˆซ0bh22dxโˆซ0b[h]dx

yยฏ=hโˆซ0bxdx2โˆซ0bxdx=hx220b2x220b=hb222b22

yยฏ=h2

Hence, centroid isxยฏ=b2,yยฏ=h2

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