# Line Integrals Examples 2

Recall from the Line Integrals page that if $z = f(x, y)$ is a two variable real-valued function and $C$ is a smooth plane curve defined by the parametric equations $x = x(t)$ and $y = y(t)$ then the line integral of $f$ along $C$ is given by:

(1)

Similarly if $z = f(x, y, z)$ is a three variable real-valued function and $C$ is a smooth space curve defined by the parametric equations $x = x(t)$, $y = y(t)$ and $z = z(t)$ then the line integral of $f$ along $C$ is given by:

(2)

We will now look at some more examples of computing line integrals.

## Example 1

**Evaluate the line integral $\int_C xyz \: ds$ where $C$ is the curve given by the parametric equations $x(t) = 2 \sin t$, $y(t) = t$, and $z = -2 \cos t$ for $0 ≤t ≤ \pi$.**

We note that $f(x,y,z) = xyz$ and so $f(x(t), y(t), z(t)) = -4t \sin t \cos t$. Furthermore, $\frac{dx}{dt} = 2 \cos t$, $\frac{dy}{dt} = 1$ and $\frac{dz}{dt} = 2 \sin t$. Thus we have that:

(3)

## Example 2

**Evaluate the line integral $\int_C z \: ds$ where $C$ is the curve given by the parametric equations $x(t) = \rho \cos t \sin t$, $y(t) = \rho \sin^2 t$, and $z(t) = \rho \cos t$ for $0 ≤ t ≤ \frac{\pi}{2}$ and $\rho > 0$.**

We note that $f(x, y, z) = z$, and so $f(x(t), y(t), z(t)) = \rho \cos t$. Furthermore, $\frac{dx}{dt} = (\rho \cos^2 – \rho \sin^2 t) = \rho \cos 2t$, $\frac{dy}{dt} = 2 \rho \sin t \cos t = \rho \sin 2t$, and $\frac{dz}{dt} = -\rho \sin t$. Thus we have that:

(4)

Now to evaluate this integral, we will need to use a few techniques of integration. We’ll start with substitution. Let $u = \sin t$ so that $du = \cos t$. Then the limits of integration become $0$ and $1$ and so:

(5)

Now to evaluate this integral, we will need to use trigonometric substitution. Let $u = \tan \theta$ so then $du = \sec^2 \theta \: d \theta$. The limits of integration become $0$ and $\frac{\pi}{4}$ and so:

(6)

Now this integral can be evaluated using integration by parts to get that $\int \sec^3 \theta \: d \theta = \frac{1}{2} \sec \theta \tan \theta + \frac{1}{2} \ln \mid \sec \theta + \tan \theta \mid$ and so:

(7)

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