How much work?
Calculus
Level
pending
How much work would it take for a bug that is infinitely small to travel from (0,0) to (2,4) on a standard (x,y) plane following the path x^2=y assuming the only "force" on the bug is the vector field F(X)=(xy,x+y).
The answer is 17.3.
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1 solution
To solve this problem we must use curve integrals. If an object, bug in this case follows a path C(t) assuming only the vector field F(X), then the work required is the integral of F(C(t))•dC/dt dt with a<=t<=b. Please note the "•" symbolizes the dot product, not multiplication. Substituting the parametrization of our curve into the integral yields: integral of F(t,t^2)•(1,2t) dt. Our vector field simplifies to (t^3, t+t^2) and therefor the dot product is simply t^3 + 2t^2 + 2t^3 or 3t^3 + 2t^2. Therefor the integral is simplifies as 3/4 t^4 + 2/3 t^3; 0<=t<=2. Evaluating this results in 52/3 or 17.3 repeating. Now rounding this answer to three significant figures yields 17.3//