1.
Compute \(\int_C zdy\) where the curve is the helix parametrized by \(\vec{r}(t)+\cos(t)\hat{i}+\sin(t)\hat{j}+t\hat{k},\ 0\leq t\leq 2\pi\)
Answer.The integral will have a short integration by parts in it and evaluates to 0
Section 2.5 Line Integrals with respect to x or y
In this video we are going to focus on line integrals with respect to just one of the variables, x or y. When our line integral is geometrically interpreted as the surface area above some planar curve beneath some function, then the line integral with respect to x or y can be interpreted as the projection of that surface onto either the xz or yz plane. Alternatively, if we are doing a Line Integral of a vector field along a curve, the line integrals with respect to dx or dy is just zeroing out the \(\hat{j}\) or \(\hat{i}\) components respectively.
Learning Objectives:
- Compute a line integral with respect to x or y
- Interpret line integrals with respect to x or y geometrically
TYPO: At 7:45 I accidentally drop the derivative primes from g' and h' despite them being present in the previous slide.
