winter_ken said:Homework Statement
Integrate some area C of (xe^y)ds where C is the arc of the curve x=e^y
Homework Equations
What is the indeffinite integral and why is it that? Answer is (1/3)e^3y + C
The Attempt at a Solution
Integral of (xe^y)((e^y)^2 + 1)^(1/2)
= Integral of (e^2y)(e^2y + 1)
A line integral is a mathematical concept used in vector calculus to calculate the value of a scalar or vector field along a curve or path.
To integrate a function in a line integral, you must first parameterize the curve or path using a variable such as t. Then, substitute the parameterization into the function and integrate with respect to the parameter.
The formula for a line integral is ∫f(x,y)ds = ∫f(x(t),y(t))|r'(t)|dt, where f(x,y) is the function being integrated, ds is the differential arc length, and r(t) is the parameterization of the curve or path.
To solve the given line integral problem, first substitute the given function into the formula for a line integral. Then, parameterize the curve or path and calculate the differential arc length. Finally, integrate the function with respect to the parameter and evaluate the integral.
Line integrals are used in many fields of science and engineering, such as physics, engineering, and economics. They are used to calculate work done by a force, electric and magnetic fields, fluid flow, and many other physical quantities.