Chapter 13: Q17E (page 805)
Find a parametric representation for the surface.
The part of hyperboloid\(4{x^2} - 4{y^2} - {z^2} = 4\) that lies in front of the xy plane.
The parametric representation of the plane is
\(\begin{array}{l}y = y\\z = z\\x = \sqrt {1 + {y^2} + \frac{1}{4}{z^2}} \end{array}\)
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Find the area of the surface with the vector equation
\({\rm{r}}(u,v) = \left\langle {{{\cos }^3}u{{\cos }^3}v,{{\sin }^3}u{{\cos }^3}v,{{\sin }^3}v} \right\rangle ,0 \le u \le \pi ,0 \le v \le 2\pi \). State your answer correct to four decimal places.
The curve with the vector equation \({\rm{r(t) = }}{{\rm{t}}^{\rm{3}}}{\rm{i + 2}}{{\rm{t}}^{\rm{3}}}{\rm{j + 3}}{{\rm{t}}^{\rm{3}}}{\rm{k}}\) is a line.
Evaluate the line integral \(\int_C {{x^2}} dx + {y^2}dy\) for the curve \(C\) which consists of the arc of the circle \({x^2} + {y^2} = 4\) from the point \((2,0)\) to \((0,2)\) followed by the line segment from the point \((0,2)\) to the point \((4,3)\).
\({\bf{F}} = |{\bf{r}}{|^2}{\bf{r}}\), where \({\bf{r}} = x{\bf{i}} + y{\bf{j}} + z{\bf{k}}\), \(S\) is the sphere with radius \(R\) and centre the origin
To find: The area of the surface that creates by the intersection of two cylinders \({y^2} + {z^2} = 1\) and \({x^2} + {z^2} = 1\).
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