Question

Evaluate the line integral with respect to arc length

Answer 1

Let C be any curve in Rd. A parametrization of C is a map γ:[a,b]→Rd which trace along the points on C in a specific order. For any two functions f, g defined on the set of points belong to C, we define the line integral over C by

∫Cfdg

def

=

∫γfdg

def

=

∫

b

a

f(γ(t))(g∘γ)′(t)dt

i.e. the line integral is defined through a integral over a specific parametrization of the curve. The key is the value of the integral on the right is independent of the choice of parametrization. For clarity, one can drop the explicit parameter t from the expression.

For your case, ∫exdx really means ∫ex(γ(t))(x∘γ)′(t)dt for whatever parametrization you choose to evaluate the integral. The s you usually see stands for the arc length parametrization, it is only one possible choice of parametrization. You don't need to use it if it make your life harder.

Answer 2

HEY HERE IS YOUR ANSWER HOPE THIS HELPS MARK AS BRAINLIST. . .

Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment is also called rectification of a curve. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases.