Find the line integral of f(x, y, z)=√(3) /(x^2+y^2+z^2) over the curve r(t)=t 𝐢+t 𝐣+t 𝐤, 1 ≤t ≤

Find the line integral of f(x, y, z)=√(3) /(x^2+y^2+z^2)...

Key Concepts

Arc Length Element

The arc length element, denoted by ds, represents an infinitesimally small segment of the curve. It is calculated as the norm of the derivative of the parameterized curve with respect to its parameter. This concept is crucial in converting the integration variable and accurately weighting the function values along the entire curve.

Improper Integral

An improper integral is an integral where either the interval of integration is unbounded or the integrand is unbounded within the interval. When the limits approach infinity or there's a singularity, special techniques are used to evaluate the integral as a limit, ensuring a proper and convergent evaluation of the integral.

Scalar Line Integral

A scalar line integral involves integrating a scalar function along a curve. The integral sums the values of the function weighted by the infinitesimal arc length ds along the path. This concept is useful in computing quantities such as mass or work that are distributed along curves.

Curve Parameterization

Parameterization of a curve is the process of expressing the coordinates of points on the curve as functions of a single parameter. This allows complex curves to be described in a simple form, which is essential for evaluating integrals along the curve by transforming the variables in the function and the differential element.

Transcript

00:01 Okay, folks, so in this video, we're going to take a look at this integration here.

00:07 We have a line integral.

00:08 This is problem number 15.

00:13 We have a line integral that we need to do.

00:15 First of all, we have the function f that we want to integrate against x, y, z.

00:21 So that's equal to x plus root y minus z squared.

00:26 So this is our function.

00:30 And we're given two space curves.

00:32 The first one is c1, and the second one is c2.

00:39 And because of the fact that, as you can probably see from the graph, c1 and c2, the point where they join is not differentiable.

00:51 The derivative has kind of a jump discontinuity there at the point 110, which is where c1 and c2 meets.

01:01 So we can't just do one integral, we have to do two, two integrals and then we're going to add them together.

01:06 So we're going to integrate the function f on the curve c1 first, and then we're going to add that to the integral of the function f along the curve c2.

01:16 So we're going to have two integrals to do instead of just one.

01:20 So let's do it.

01:21 So for c1, we have the space curve r is equal to ti, which is, i'm going to write i as x -hat, because that's the same thing, plus t squared in the y hat direction.

01:40 And then t is from zero to one.

01:44 Okay, so that's for c1.

01:46 What about c2? well, c2 is just one x hat plus one j hat plus, let's see what this is, plus t, z hat.

02:01 Okay, and then the limits of integration is also from all the way up to one.

02:10 Okay, i realized i missed something here.

02:12 I missed the t.

02:13 All right.

02:14 So now we have been given the two space curves that we want to integrate along.

02:19 So let's do the first integral.

02:20 Let's do the integral for c1.

02:25 And then we have c1, f as a function of x, y, and z multiplied by d .s, plus c2 of f as a function of x, y, and z, multiplied by d .s, we're going to calculate these two integrals separately, and then we're going to add them together.

02:45 Okay, so that's our strategy for this one.

02:48 We have, now i'm going to rewrite.

02:52 What i'm going to do is i'm going to rewrite x, y, and z, all of them as a function of t.

02:57 So as you can see, along on the curve c1, x as a function of t, if you look here, it's just t.

03:05 Okay, so we have t plus the square root of t squared, which is going to give you you know, which is going to give you the absolute value of t, but because of the fact that we're restricting ourselves to only the points where t is bigger than zero, so the absolute value of t in this range is really just t itself, that's kind of a triviality that you don't need to pay too much attention to.

03:32 So we have square root of t squared is going to give you t, and then z is zero.

03:40 So i'm going to leave this term, alone, multiplied against ds, but what's ds? well, ds is really just, you know, x prime squared plus y prime squared, plus z prime squared, multiplied by dt, and then what this is going to give you is, let's calculate t prime, i mean, x prime, well, x is t, so x prime is 1, 1 squared is this one, plus, well, y is a function of t is t squared.

04:14 So the derivative of that is going to give you 2t squared, plus there's no z, so the last part is 0, multiplied by dt.

04:31 Okay, so this is our first curve, and what we're going to do, let's figure out what the limits of integration is.

04:38 Well, we're going from zero to one.

04:42 Okay, so that's for the first one.

04:43 Let's see what this is...

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