Line integral question, answer is here, just confused on it

In summary, the conversation discusses a line integral over C and the confusion surrounding the substitution and resulting integrand. It is determined that there is a typo in the original equation and that the 2t dt term is equal to the differential dz.
  • #1
mr_coffee
1,629
1
Hello everyone I'm confused on this line integral.
The substiution is easy but I'm not sure where 2t is coming from...

integral over C x^2*y*sqrt(z) dz;
C: x = t^3;
y = t;
z = t^2;

0 <= t <= 1

integral over C x^2*y*sqrt(z) dz =
integral 0 to 1 (t^3)^2 (t) sqrt(t) * 2t dt =
integral 0 to 1 2*t^9 dt;

Okay I see they are just plugging in the t's for the x,y,z, but why do they write sqrt(t)? the sqrt(t^2) is just t.

Also where is this 2t dt coming from?

I thought maybe they used: sqrt(t^2); u = t^2;
du = 2t dt;
1/2*du = t dt;


Thanks
 
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  • #2
I guess sqrt(t) is just a typo. It should be sqrt(t^2) (Note that the final integral is t^9). And that 2tdt is equal to the differential dz.
 
  • #3
What Neutrino said. The final integrand should be 2t^9. In fact, if you compare step 2 to step 3, you'll find that it does not logically follow. So in all likelihood it was a typo.
 
  • #4
THanks for the help guys!
 

1. What is a line integral?

A line integral is a type of integral used in multivariable calculus to calculate the total value of a function along a given path or curve. It takes into account both the magnitude and direction of the function along the path.

2. How is a line integral different from a regular integral?

A regular integral involves integrating a function over a single variable, typically denoted as x. A line integral involves integrating a function over a curve or path in a multi-dimensional space, typically denoted as . Additionally, a line integral takes into account the direction of the function along the curve, whereas a regular integral does not.

3. When is a line integral used?

A line integral is used in various fields such as physics, engineering, and mathematics to calculate quantities such as work, electric potential, and fluid flow. It is also commonly used in vector calculus to solve problems involving surfaces and curves.

4. How is a line integral calculated?

To calculate a line integral, you first need to parameterize the curve or path in terms of a variable, typically t. Then, you evaluate the integral by plugging in the parameterized equation into the function and integrating over the specified limits. The final result is a single value representing the total value of the function along the given curve.

5. What are some practical applications of line integrals?

Some practical applications of line integrals include calculating the work done by a force along a given path, finding the electric potential along a wire or circuit, and calculating the amount of fluid flowing through a pipe. Additionally, line integrals can be used to find the length and surface area of curves and surfaces respectively.

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