Find the line integral of f(x, y)=y e^x^2 along the curve r(t)=4 i-3 t j,-1 ≤t ≤2.

Name: Problem 21, Chapter 16: Thomas' Calculus, Multivariable (12th Edition)
Uploaded: 2020-05-30T00:16:27-08:00
Duration: 6 min 21 s
Channel: Jack Hou
Description: Find the line integral of f(x, y)=y e^x^2 along the curve r(t)=4 i-3 t j,-1 ≤t ≤2.

Find the line integral of f(x, y)=y e^x^2 along the curve...

Key Concepts

Differential Element Transformation

Transforming the differential element entails replacing the differential in the original variables with a differential in the parameter. When the line integral is taken over the parameterized curve, this often includes multiplying by the derivative’s magnitude when the integration is with respect to arc length, or directly using the derivative if the integral is set up with respect to the parameter.

Function Composition for Integration

Function composition in the context of integration refers to composing the given scalar function with the parameterization of the curve. This allows the evaluation of the function along the curve by expressing all variables in terms of the parameter.

Curve Parameterization

Parameterizing a curve means representing it by a vector function of a parameter, which allows the conversion of a line integral into a standard integral with respect to that parameter. This process involves substituting the curve's coordinate expressions into the function being integrated, thereby simplifying the evaluation.

Line Integral of a Scalar Field

A line integral of a scalar field involves integrating a function along a curve where the function’s values are accumulated. This method is often used to compute quantities like mass or work along a path, and it involves expressing the scalar function in terms of a parameter that describes the curve.

Transcript

00:01 Okay, folks, so in this video, we're going to be talking about problem number 21 on your book.

00:07 We have a line integral to solve.

00:11 We have the first of all, we have the function, which is a function of x and y.

00:15 And that function is y multiplied by e to the power of x squared.

00:23 That's weird.

00:25 And then we're given a curve, which is parameterized by the parameter t.

00:31 And that curve is right here from this point all the way down to this point.

00:39 And of course, i did a little bit of work beforehand.

00:42 And, you know, what you're actually given in the book is this.

00:45 We have r as a function of t is 4ti minus 3tj.

00:52 Okay.

00:52 And then i took this thing and then rewrote it like this because that makes it a little bit simpler to analyze.

01:01 Because if you write it in this form, you can see that this is really just a vector.

01:06 If you first write out the vector 4 -93, which is somewhere here, 4 -9 -3, and then you let the variable t vary, you're going to get a curve, a line segment to be precise, that looks exactly like this.

01:25 Okay, so we're going to evaluate the function f along the curve.

01:32 We're going to integrate it along the curve, so let's write it out.

01:36 We have integral along the curve r, y, e, x squared, multiplied by d .s, of course.

01:45 But d .s, we're going to rewrite the s in another way, which is the usual way of rewriting the infinitesimal line segment.

01:54 We have x as a function of t is 4t, so x prime is 4, and 4 squared is 16.

02:02 So we have 16 plus y prime while y as a function of t is negative 3 t y prime is negative 3 okay so y prime squared gives you 9 multiplied by d t okay so i'm going to erase this and then i have a square root of 25 multiplied by d t um but that is a constant so i can pull it out of the integral we have we have five because that's what squared of is multiplied by y which is negative 3t multiplied by e to the power of x squared but x is a function of t and that function is 4t so x squared gives you 16 t squared multiplied by d t now we can try to figure out what the the limits of integration is well we started off from x equals 4 t which we started we started off from this point, which is when 4, which is when t equals, let me see what problem is this.

03:11 This is problem number 21, and when t is negative 1, this is our starting point, this is our starting position, x equals 4t, and let's see here.

03:28 I'm having a little bit of a problem here...

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