# 16.1.9 Line Integral over space curves

• MHB
• karush
In summary, a line integral over space curves is a mathematical concept used to calculate the integral of a scalar or vector function along a three-dimensional curve in space. It differs from a regular line integral in that it is calculated along a three-dimensional curve and the direction of the curve is significant. Real-world applications include calculating work, flux, and center of mass. To solve a line integral over space curves, one needs a strong understanding of vector calculus and operations such as dot and cross products.
karush
Gold Member
MHB
Evaluate

$\displaystyle \int_C(x+y)ds$
where C is the straight-line segment
$x=t, y=(1-t), z=0,$
from (0,1,0) to (1,0,0)

ok this is due tuesday but i missed the lecture on it
so kinda clueless.
i am sure it is a easy one.

Use $ds = \sqrt{(dx/dt)^2+(dy/dt)^2+(dz/dt)^2}dt$, take derivative of $(x(t),y(t),z(t))$ with respect to t, and then the integral goes from $(x,y,z)=(0,1,0)$ to $(1,0,0)$ check what it means for the t variable.

## 1. What is a line integral over space curves?

A line integral over space curves is a mathematical concept that involves calculating the integral of a scalar or vector function along a curve in three-dimensional space. It is used to determine the total amount of a quantity, such as work or mass, that is accumulated along a path in space.

## 2. How is a line integral over space curves different from a regular line integral?

A line integral over space curves differs from a regular line integral because it is calculated along a three-dimensional curve in space, rather than a two-dimensional curve in the xy-plane. This means that the curve must be parameterized in terms of three variables, usually x, y, and z.

## 3. What is the significance of the direction of the curve in a line integral over space curves?

The direction of the curve in a line integral over space curves is important because it determines the orientation of the integral. Reversing the direction of the curve will result in a negative value for the line integral, while keeping the direction the same will result in a positive value.

## 4. What are some real-world applications of line integrals over space curves?

Line integrals over space curves have numerous applications in physics and engineering, including calculating work done by a force along a curved path, determining the flux of a vector field through a curved surface, and finding the center of mass of a three-dimensional object.

## 5. What mathematical tools are needed to solve a line integral over space curves?

To solve a line integral over space curves, one needs a strong understanding of vector calculus, including vector fields, line integrals, and multiple integrals. Knowledge of parameterization and vector operations, such as dot and cross products, is also necessary.

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