example
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EX1 Find the slope of the tangent to the asteroid x=acos3θ, y=asin3θ as a function of the parameter θ
- dxdy=x′(θ)y′(θ)=−3acos3θsinθ3asin3θcosθ
- ⇒ −tanθ
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At what point is the tangent is horizontal
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At what point is the tangent vertical
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at what point is slope = 1
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at what point is slove equal to -1
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Ex 2, find the area under on arch of the cycloid
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Area
- x=r(t−sint),y=r(1−cost),0≤t≤2π
- A=∫02πy(x)dx=∫02πy(t)x′(t)dx
- /∫02πr(1−cost)r(1−cot)dt
- ⇒ r2∫02π1−2cost+cos2tdt
- ⇒ r2(t−2sint+21t+41sin2t)∣02π
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Arc length
- L=∫02πx′(t)2+y′(t)2dt ⇒ ∫02πr2(1−cost)2+r2sin2tdt
- ⇒ r∫02π1−2cost+cos2t+sin2tdt ⇒ r∫02π2−2costdt
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Ex 3 Find the surface area
- x=3t−t3, y=3t2,0≤t≤1
- A=∫012π3t2(3−3t2)2+(6t)2dt
- 6π∫01t29+9t4−18t2+36t2dt
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Ex 4 Polar coordinate are parametric equastions
- r=1+cosθ
- x′=r′(θ)cosθ−r(θ)sinθ
- y′=r′(θ)sinθ+r(θ)cosθ
- (r′cosθ−rsinθ)2+(r′sinθ+rcosθ)2dθ
- ⇒ r′cos2θ+r2sin2θ−2rr′cosθsinθ+r′2sin2θ+r2cos2θ+
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finding the area using the equasion
- L=∫02π(sin2θ+(1+cosθ)2)dθ
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finding the volume of revolution
- A=∫0π2πr(θ)sinθ2+2cosθdθ
- ⇒ 2π∫0π(1+cosθ)sinθ2+2cosθdθ