armolinasf said:Homework Statement
Evaluate the line integral:
int(ydx+xdy) where the path C is y=sinx from (0,0) to (pi/2,0)
Homework Equations
The Attempt at a Solution
(pi/2,0) is not a solution to y=sinx. I could use the fundamental theorem but for my potential function I get F(x,y)=xy which would give me zero with those endpoints...answer in the book gives pi/2. Is this an error? Thanks
A line integral is a type of integral that is used to calculate the total value of a function along a specified curve or path in a two- or three-dimensional space. It is typically denoted by ∫Cf(x,y) ds, where C is the curve and f(x,y) is the function being integrated.
A regular integral is used to find the area under a curve, while a line integral is used to find the total value of a function along a specified curve or path. In a regular integral, the independent variable is usually x, while in a line integral, the independent variable is typically the arc length of the curve. Additionally, in a line integral, the limits of integration are defined by the starting and ending points of the curve.
A closed line integral is one in which the starting and ending points of the curve are the same. These types of integrals are important in physics, as they can be used to calculate quantities such as work, flux, and circulation. In mathematics, closed line integrals are also used in the study of vector fields and their properties.
To calculate a line integral, you first need to parameterize the curve or path in terms of a single variable, such as t. Then, substitute the parameterized equation into the integral, and integrate with respect to the parameter t. The result will be a scalar value representing the total value of the function along the specified curve.
One common error is not properly parameterizing the curve, which can lead to incorrect results. Another error is not taking into account the direction of the curve, which can result in a negative sign in the final answer. It is also important to correctly identify the limits of integration and to use the correct formula for the type of line integral (e.g. closed vs open).