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Area enclosed by star line x = ACOS ^ 3T, y = asin ^ 3T
From symmetry, s = 4 ∫ (0 → a) YDX = 4 ∫ (π / 2 → 0) a (Sint) ^ 3 D [a (cost) ^ 3] = 12a ^ 2 × ∫(0→π/2) (sint)^4 × (cost)^2 dt＝12a^2 × ∫(0→π/2) [(sint)^4－(sint)^6] dt＝12a^2 × [3/4 × 1/2 × π/2－5/6 × 3/4 × 1/2 × π/2]＝(3πa...
Differential problem, find the equation of tangent L (T) of curve y = ACOS't, x = asin't at point (x (T), y (T))
Slope of tangent l k = dy / DX = (dy / DT) / (DX / DT) = - a Sint) / (a cost) = - tant
Substitute points (x (T), y (T)) into
y=(-tant)*(asint)+acost
Find the curve; x=acos^3 Θ: y=asin^3 Θ , stay Θ= Tangent equation at π / 6, normal equation,
dy/dx=(dy/d Θ)/ (dx/d Θ)= (3asin^2 Θ cos Θ)/ (-3acos^2 Θ sin Θ)=- tan Θ Dy (π / 6) / DX = - root 3 / 3x (π / 6) = 3A root 3 / 8 y (π / 6) = A / 8 tangent equation: Y-A / 8 = - root 3 / 3 * (x-3a root 3 / 8) y = - root 3 / 3 * x + A / 2 normal equation: Y-A / 8 = root 3 * (x-3a root 3
Find the surface area of the surface formed by the rotation of the curve X = acos3t, y = asin3t (a > 0) around the straight line y = X
This is a star line. Let the resulting surface of rotation be Σ and the surface area of Σ be a
Fully considering the symmetry (x = 0, y = 0, x = y, x = - y), there are
The surface area a is twice the surface area obtained by rotating the curve around y = x in the first quadrant
‡ from the area formula of the rotating surface
A=2∫
3π
four
π
four
2π|x-y|
two
x′2+y′2dt
=6
2πa2[∫
π
two
π
four
(-cos3t+sin3t)sintcostt+∫
three
π4
π
two
(cos3t-sin3t)sintcostt]
=6
2πa22-2(1
2)5
5=12
2-3
5πa2
If x = 1 / 2T, y = 1 / 3T, x-6y + 3 / 4 = 0, then t=
x=1/2t,y=1/3t,x-6y+3/4=0
arcsinx-x
1/2 t - 2 t +3/4=0
The solution is t = 1 / 2
Solving equations (1) y = x + 3 7x + 5Y = 9 (2) 3s-t = 5 5S + 2T = 15 (3) 3x + 4Y = 16 5x-6y = 33 (4) 4 (x-y-1) = 3 (1-y) - 2 / 2 x + 3 / 3 y = 2
1.7x + 5 (x + 3) = 9 7x + 5x + 15 = 9 x = - 0.5 (- 1 / 2) substitute x into y = x + 3, y = 2.52.3s-5 = t 5S + 2 (3s-5) = 15 11s-10 = 15 s = 25 / 11 substitute x into 3s-5 = t, t = 1 and 93.3x + 4Y = 16 of 11 deform into x = (16-4y) / 3 into 5x-6y = 33, 5 * (16-4y) / 3-6y = 33, y = - 1 / 2
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