Line Integrals with Respect to Specific Variables Examples 1
Recall from the Line Integrals with Respect to Specific Variables page that if $z = f(x, y)$ is a two variable real-valued function and $C$ is a smooth plane curve parameterized as $x = x(t)$ and $y = y(t)$ for $a ≤ t ≤ b$ then the line integral of $f$ along $C$ with respect to $x$ and the line integral of $f$ along $C$ with respect to $y$ are:
When these integrals occur together, we also use the shortened notation:
The same notation and definitions are used for line integrals with respect to $x$, $y$, or $z$ of the three variable real-valued function $w = f(x, y, z)$.
Let’s now look at some examples of computing these sort of integrals.
Evaluate $\int_C z \: dx + x \: dy + y \: dz$ where $C$ is the curve given parametrically by $x = t^2$, $y = t^3$, and $z = t^2$ for $0 ≤ t ≤ 1$.
We note that $x'(t) = 2t$, $y'(t) = 3t^2$ and $z'(t) = 2t$, and so, using the formula directly and we have that:
Evaluate $\int_C z^2 \: dx + x^2 \: dy + y^2 \: dz$ where $C$ is the line segment that joins the points $(1, 0, 0)$ and $(4, 1, 2)$.
We need to first parameterize this line segment. Fortunately, this is easy to do. For $0 ≤ t ≤ 1$ we will have that:
Thus we have that $x(t) = 1 + 3t$, $y(t) = t$ and $z(t) = 2t$. Therefore: