Scalar Line Integrals

Calculus 3: Line Integrals (18 of 44) What is a Line Integral? [(y)dx+(z)dy+(x)dz] Example 6
Calculus 3: Line Integrals (18 of 44) What is a Line Integral? [(y)dx+(z)dy+(x)dz] Example 6

1.
Know what a scalar line integral is and what it represents geometrically.
2.
Understand the scaling factor when computing a scalar line integral.
3.
Be able to compute scalar line integrals.

Recap and Understanding

First, take a look at this recap video going over the basics of scalar line integrals.

Important things out of the video:

  • We have a procedure for computing scalar line integrals:
  • Geometrically, the scalar line integral of any function over a curve
    represents what? The arc length of the curve The signed area under the function above the curve
  • We can get the arc length of a finite curve by taking the scalar line integral over of the function .

Example Video

Take a look at the following video working through the following problem:

The other way to look at it is to say that bends and stretches a time interval into the curve . The is giving you how much you are stretching the interval, which will consequently scale your area by the same amount.

Problems

  • We first parametrize . For this curve, we could let , in which case and our parametrization is:
  • The bounds for when we do this: must be in the interval .
  • In this case,
  • We also compute : we get , and so
  • The integral can therefore be written as:
  • Evaluating this integral gives an answer of

As we will see in this next problem, the process is exactly the same for functions of three variables. Just apply the formula in the same way, but now your parametrization will have three components instead of two (but still only one parameter!).

True/False

Here are some true/false questions to get you thinking a bit more about scalar line integrals.

You are watching: Scalar Line Integrals. Info created by THVinhTuy selection and synthesis along with other related topics.

Rate this post

Related Posts