Scalar Line Integral

Section 3.2 Scalar Line Integral

Subsection 3.2.1 Coordinate System Definitions

Surfaces consist of many curves. Section 2.1 and Section 2.4 presented the concept of many curves passing through a surface and how this applies to limits and derivatives. One type of integral is also defined in terms of the curves of a surface. Once again we extend single-variable calculus to obtain a result in multi-variable calculus.

Subsection 3.2.2 Derivation

Activity 50.

In this activity we illustrate a type of integral, by using curves in a surface to convert the problem into a single variable problem.

Use the function \(h(x,y)=200-x^2-y^2\) for the following.

(a)

Example 1: use the path \(p(t)=(t,0)\text{.}\)

(i)

Write \(f(t)=h(p(t))\text{.}\)

(ii)

Calculate \(p(t)\) for \(t=0,1,2,3\text{.}\)

(iii)

What is the distance between each of these points?

(iv)

Calculate \(h(p(t))\) for \(t=0,1,2,3\text{.}\)

(v)

Use the distances between \(p(t)\) and the heights from \(h(p(t))\) to find the area of rectangles (think Riemann integration). What is the total, approximate area?

(vi)

Now setup an integral to calculate the exact area between this curve and the \(xy\)-plane.

(b)

Example 2: use the path \(p(t)=(0,t)\text{.}\)

(i)

Write \(f(t)=h(p(t))\text{.}\)

(ii)

Calculate \(p(t)\) for \(t=0,1,2,3\text{.}\)

(iii)

What is the distance between each of these points?

(iv)

Calculate \(h(p(t))\) for \(t=0,1,2,3\text{.}\)

(v)

Use the distances between \(p(t)\) and the heights from \(h(p(t))\) to find the area of rectangles (think Riemann integration). What is the total, approximate area?

(vi)

Now setup an integral to calculate the exact area between this curve and the \(xy\)-plane.

(c)

Example 3: use the path \(p(t)=(t,t)\text{.}\)

(i)

Write \(f(t)=h(p(t))\text{.}\)

(ii)

Calculate \(p(t)\) for \(t=0,1,2,3\text{.}\)

(iii)

What is the distance between each of these points?

(iv)

Calculate \(h(p(t))\) for \(t=0,1,2,3\text{.}\)

(v)

Use the distances between \(p(t)\) and the heights from \(h(p(t))\) to find the area of rectangles (think Riemann integration). What is the total, approximate area?

(vi)

Now setup an integral to calculate the exact area between this curve and the \(xy\)-plane.

(d)

Example 4: use the path \(p(t)=(cos(t),sin(t))\text{.}\)

(i)

Write \(f(t)=h(p(t))\text{.}\)

(ii)

Calculate \(p(t)\) for \(t=0,\pi/2,\pi,3\pi/2\text{.}\)

(iii)

What is the distance along the curve between pairs of these points?

(iv)

Calculate \(h(p(t))\) for \(t=0,\pi/2,\pi,3\pi/2\text{.}\)

(v)

Use the distances between \(p(t)\) and the heights from \(h(p(t))\) to find the area of rectangles (think Riemann integration). What is the total, approximate area?

(vi)

Now setup an integral to calculate the exact area between this curve and the \(xy\)-plane.

Subsection 3.2.3 Method

Definition 3.2.1. Scalar Line Integral.

For a a smooth, parameterized curve \(C\) given by \((x(t),y(t))\) for \(t \in [a,b]\) the scalar line integral of a function \(f:\R^2 \to \R\) is

\begin{equation*} \int_C f(x,y) \; ds = \lim_{\Delta s \to 0} \sum_{i=0}^n f(x^*,y^*) \Delta s. \end{equation*}

For a continuous function \(f\) the scalar line integral can be calculated using

\begin{equation*} \int_C f(x,y) \; ds = \int_a^b f(x(t),y(t)) \sqrt{(x^\prime)^2+(y^\prime)^2} \; dt. \end{equation*}

Figure 3.2.2. Illustration of Scalar Line Integral

Exercises 3.2.4 Exercises

1.

Integrate \(f(x,y)=\sin(x+y)\) along the curve \(y=4x-1\) for \(x \in [0,5].\)

2.

Integrate \(f(x,y)=5x+3y+7\) along the unit circle.

3.

Integrate \(f(x,y)=x^3\) along the curve \(y=x^3\) for \(x \in [-1,1].\)

4.

Integrate \(f(x,y)=x\sin y\) along the triangle with vertices \((0,0),\) \((3,0),\) \((0,4).\)