Line integral question

• asdf1
In summary, the line integral of the function v=(y的平方, 2x(y+1), 0) from the point a=(1,0,0) to the point b=(2,2,0) is 11. However, the parameter equations x=1+t, y=1+t, z=0 do not accurately represent the path from a to b. A more accurate parametrization would be x=t+1, y=2t, z=0. When t=0, this parametrization gives the point (1,0,0) which

asdf1

Calculate the line integral of the function v=(y的平方, 2x(y+1), 0) from the
point a=(1,0,0) to the point b=(2,2,0)
The correct Ans: 11

However, when I was calculuating the problem,
I supposed(is that the right word?) to make the parameter equations,

x=1+t
y=1+t
z=0
where 0<t<1
but when I integrated the function, the answer is 6.

i don't know why, but i see your function as some chinese characters, can you retype it in some other way:).

I'd say

$$\left\{ \begin{array}{l} x = t \\ y = 2t - 2 \\ z = 0 \\ \end{array} \right$$

or

$$\left\{ \begin{array}{l} x = t + 1 \\ y = 2t \\ z = 0 \\ \end{array} \right$$

asdf1 said:
Calculate the line integral of the function v=(y的平方, 2x(y+1), 0) from the
point a=(1,0,0) to the point b=(2,2,0)
The correct Ans: 11

However, when I was calculuating the problem,
I supposed(is that the right word?) to make the parameter equations,

x=1+t
y=1+t
z=0
where 0<t<1
but when I integrated the function, the answer is 6.

when t= 0, your parametrization gives (1, 1, 0), there is no value of t that gives (1, 0, 0).

sorry, i mistyped~
a=(1,0,0)

1. What is a line integral?

A line integral is a mathematical concept used in vector calculus to calculate the total value of a scalar or vector field along a given curve or path. It is represented by a definite integral and is often used to solve problems in physics and engineering.

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

A line integral is different from a regular integral in that it calculates the value of a field along a specific curve, rather than over a two-dimensional region. It also takes into account the direction of the curve, as it is a path-dependent integral.

3. What is the purpose of a line integral?

The purpose of a line integral is to calculate the total value of a field along a specific curve or path. This is useful in many applications, such as calculating work done by a force, finding the mass of a wire, or determining the flow of a fluid through a curved pipe.

4. What are some real-world applications of line integrals?

Line integrals have many real-world applications in physics, engineering, and other sciences. Some examples include calculating electric potential and work done in electric circuits, determining the mass of a wire or cable, and finding the fluid flow rate in a curved pipe.

5. How do you calculate a line integral?

To calculate a line integral, you first need to parameterize the curve or path along which the integral will be evaluated. Then, you need to set up the integral using the parameterization and the function for the field. Finally, you can solve the integral to find the total value of the field along the given curve.