Curves and line integrals of scalar functions

1 Curves and line integrals of scalar functions

Find the curve of intersection of the cylinder \((x - 3) ^2 + (z + 1 )^2 = 9\) and the plane \(2x + 3y - 2z = 5\)
First find \(x(t)\) and \(z(t)\) so that no matter how you define \(y(t)\), the curve you get lives in the cylinder and wraps around once.
Find parametrizations of the curves that form the sides of the triangle \(\Sigma \), where \(\Sigma \) is the intersection of the plane \(- x - 2y + z = 7 \) with the first octant.
For points \(p\) and \(q\), the line segment from \(p\) to \(q\) can be parametrized as \(\gamma (t) = p + t (q - p)\) with \(0 \leq t \leq 1\).
Find parametrizations of the curves that form the edges of the surface \(\Sigma \), where \(\Sigma \) is the portion of the sphere \( (x- 3) ^2 + (y - 2 ) ^2 + (z- 5 ) ^2 = 36 \) with \(x \geq 3\) and \(z \geq 5\).
\(\Sigma \) is like the peel of an orange slice. The edges are two half-circles going from \((3, -4, 5)\) to \((3,8,5)\)
Determine whether the curve
\[ \gamma (t) = ( t \sin t , t^2 -1 , 3t + \cos t ) \]
passes through the point \((\pi / 2 , \pi ^2 /4 - 1 , 3 \pi / 2 )\). If it does, determine
  • the times at which it does
  • the lines tangent to the curve at those times
  • the speed of the curve at those times
Determine whether the curve
\[ \gamma (t) = ( t ^2 - 2 , t^3 -2 , 5t + t^2 ) \]
passes through the point \(( 7 , 25, 24 )\). If it does, determine
  • the times at which it does
  • the lines tangent to the curve at those times
  • the speed of the curve at those times
Determine whether the curve
\[ \gamma (t) = ( 5 + t , t^2 -1 , 3 t^2 - 2t ) \]
passes through the point \(( 7 , 2 , 8 )\). If it does, determine
  • the times at which it does
  • the lines tangent to the curve at those times
  • the speed of the curve at those times
Determine whether the curve
\[ \gamma (t) = ( t^2 + 2t +1 , t^3 + 3t^2 - t + 2 , t^3 - 7t + 3 ) \]
passes through the point \(( 4 , 5 , -3 )\). If it does, determine
  • the times at which it does
  • the lines tangent to the curve at those times
  • the speed of the curve at those times
Find the length of the curve
\[ \gamma (t) = ( e^t \cos t , e^t \sin t ) \]
with \(0 \leq t \leq 8 \pi \). Draw a sketch of the curve.
The length of a curve is the integral of the speed \(\vert \gamma ' (t) \vert \) from initial time to final time.
Find the length of the curve
\[ \gamma (t) = ( t - \sin t , 1 - \cos t ) \]
with \(0 \leq t \leq 4 \pi \). Draw a sketch of the curve.
The length of a curve is the integral of the speed \(\vert \gamma ' (t) \vert \) from initial time to final time.
Find the length of the curve
\[ \gamma (t) = ( 2 t , e^t + e ^{-t} ) \]
with \( - \log 2 \leq t \leq \log 2 \). Draw a sketch of the curve.
The length of a curve is the integral of the speed \(\vert \gamma ' (t) \vert \) from initial time to final time.
Compute the line integral
\[ \int _C ( x + 3y + z^2 ) \, ds \]
where \(C\) is the curve with parametrization \(\gamma : [0, 2] \to \mathbb {R}^3\) given by
\[ \gamma (t) = ( t-2 , 2t + 3, 5 - t ) \]
The parametrization of a curve can be used as a change of variables. In this case, \(x = t-2\), \(y = 2t + 3\), \(z = 5-t\). This integral is computed using this change of variables with the Jacobian \(\vert \gamma ' (t) \vert \).
Compute the line integral
\[ \int _C ( -8 x + 3 y - 5 z ) \, ds \]
where \(C\) is the curve with parametrization \(\gamma : [-1, 3] \to \mathbb {R}^3\) given by
\[ \gamma (t) = ( t^2 - 4 , t^2 - 2 t - 5 , 3 - t^2 ) \]
The parametrization of a curve can be used as a change of variables. In this case, \(x = t^2-4\), \(y = t^2 -2t-5\), \(z = 3 - t^2\). This integral is computed using this change of variables with the Jacobian \(\vert \gamma ' (t) \vert \).
Compute the line integral
\[ \int _C ( x + y + z ) \, ds \]
where \(C\) is the curve with parametrization \(\gamma : [ 0 , 3 \pi /2 ] \to \mathbb {R}^3\) given by
\[ \gamma (t) = ( \cos t , \sin t, t ) \]
The parametrization of a curve can be used as a change of variables. In this case, \(x = \cos t\), \(y = \sin t\), \(z = t\). This integral is computed using this change of variables with the Jacobian \(\vert \gamma ' (t) \vert \).
Compute the line integral
\[ \int _C x \, ds \]
where \(C\) is the portion of the parabola \(y = x^2 + 1 \) between \((0,1)\) and \((2,5)\).
First find a parametrization of the curve. Then compute like the previous problems.
Compute the line integral
\[ \int _C x e ^y \, ds \]
where \(C\) is the line segment from \((0,2)\) to \((5,0)\).
First find a parametrization of the curve. Then compute like the previous problems.
Compute the line integral
\[ \int _C \sqrt { 2 + 4x + 8y } \, ds \]
where \(C\) is the portion of the curve of intersection of the cylinder \(y = x^2\) and the plane \(x+ y+ z = 0\) from \((0,0,0)\) to \((2,4,-6)\)
First find a parametrization of the curve. Then compute like the previous problems.