Evaluate the line integral Integral of C= ydx+x2dy where C=C1+C2. C1 is the path
ID: 3193297 • Letter: E
Question
Evaluate the line integral Integral of C= ydx+x2dy where C=C1+C2. C1 is the path of the straight line segment from the origin, (0, 0) to the point (2, 18). C2 is the path of the parabola y=?x^2+9x+4 from the point (2, 18) to the point (5, 24) . Will rate! Thanks!Explanation / Answer
Line Integrals and Vector Fields The origin of the notion of line integral (really a path integral) comes from the physical notion of work. We will see that particular application presently. Our ?rst task is to give a de?nition of what a path and line integrals are and see some examples of how to compute them. 1 Path Integrals We start with a de?nition; later on we will try to justify the de?nition in terms of appropriate Riemann sums. The problem that we want to treat is that of integrating a real-valued function of two or three variables along a curve in two- or three-dimensional space. We will give our de?nitions in terms of R 3 . We start with a given a function f de?ned on a domain D ? R 3 so that f : D -? R, and a curve c with parameter domain an interval of the real line, c : [a, b] -? D, c(t) = (x(t), y(t), z(t)) > , a = t = b. Since for all t ? [a, b], c(t) ? D, we can form the composition f ? c so that (f ? c)(t) = f(x(t), y(t), z(t)), a = t = b. The path integral of the function f along the curve c is then de?ned as the integral of the scalar-valued function Rb a f(x(t), y(t), z(t)) kc 0 (t)k dt. (1) Notice that the factor kc 0 (t)k is the speed of traversal of the curve as t runs between endpoints of the interval a and b. To give a simple interpretation in a special case, recall that, if f = 1, then the integral of a function f over a domain D ? R 2 , ZZ D f(x, y) dx dy = ZZ D 1 dx dy , 1gives just the area of the domain D. In the present case of the pat
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.