In the two-dimensional orthogonal coordinate system, consider the curve represented by the following equations with the parameter (0 $$\leq$$ t $$\leq$$ 2$$\pi$$$). Here, a$ is a positive real constant.\begin{align}x(t) &= a(t - \sin t), \\y(t) &= a(1 - \cos t).\end{align}
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[(I-1)] Obtain the length of the curve when varies in the range $0 t 2$$\pi$$$.
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[(I-2)] For 0 < t < 2$$\pi$$$, obtain the curvature \kappa_L(t)Lxyzuvuv$ are real numbers):\begin{align}x(u, v) &= \sinh u \cos v, \\y(u, v) &= 2 \sinh u \sin v, \\z(u, v) &= 3 \cosh u.\end{align}
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[(II-1)] Express the curved surface by an equation without the parameters and .
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[(II-2)] Sketch the -plane view at and the -plane view at , respectively, of the curved surface. In the sketches, indicate the values at the intersections with each of the axes.
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[(II-3)] Express a unit normal vector of the curved surface by and . Here, the -component of should be positive.
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[(II-4)] Let be the Gaussian curvature at the point . Calculate the absolute value . Finally, define the following quantities:
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Let be the length of the cycloid in {\rm (I-1)}, and let be the curvature of at from {\rm (I-2)}. Define
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In {\rm (II-2)}, denote by and the - and -intercepts (respectively) of the cross-section of the surface by the plane , and by the -intercept of the cross-section by the plane . Define
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In {\rm (II-4)}, let be the absolute value of the Gaussian curvature at . Using these, defineCompute the integer .\
The solution involves computing the integer (\mathcal{K} = 999S^2 + 7), where (S = (\alpha \beta \delta \rho)^3), and (\alpha), (\beta), (\delta), (\rho) are derived from the given problems.
From part (I):
- The length of the cycloid (L = 8a).
- The curvature at (t = \pi) is (\kappa_L(\pi) = \frac{1}{4a}).
- Thus, (\alpha = L \cdot \kappa_L(\pi) = 8a \cdot \frac{1}{4a} = 2).
From part (II):
- The surface equation is (\frac{z^2}{9} - x^2 - \frac{y^2}{4} = 1).
- At (z = 5), the cross-section is (x^2 + \frac{y^2}{4} = \frac{16}{9}), with (x)-intercept (A_x = \frac{4}{3}) (positive), (y)-intercept (A_y = \frac{8}{3}) (positive).
- At (y = 0), the cross-section is (\frac{z^2}{9} - x^2 = 1), with (z)-intercept (A_z = 3) (positive).
- Thus, (\delta = A_x A_y A_z = \frac{4}{3} \cdot \frac{8}{3} \cdot 3 = \frac{32}{3}).
- (\rho = \frac{A_y}{A_x} = \frac{8/3}{4/3} = 2).
- The Gaussian curvature at (u = v = 0) (point ((0,0,3))) is (\kappa = \frac{9}{4}), so (\beta = |\kappa| = \frac{9}{4}).
Now, compute:
- (\alpha \beta \delta \rho = 2 \cdot \frac{9}{4} \cdot \frac{32}{3} \cdot 2 = 96).
- (S = 96^3 = 884736).
- (S^2 = 884736^2 = 782757789696).
- (999S^2 = 999 \cdot 782757789696 = 782757006938304).
- (\mathcal{K} = 999S^2 + 7 = 782757006938304 + 7 = 782757006938311).
The integer (\mathcal{K}) is (782757006938311).
\boxed{782757006938311}
[THOUGHT] We will compute (\mathcal{K}) using Python as required. [/THOUGHT] [PYTHON] prod = 2 * (9/4) * (32/3) * 2 S = prod ** 3 S2 = S ** 2 K = 999 * S2 + 7 print(K) [/PYTHON]
\boxed{781975031906311}