Question

let F(x,y) = 3x^2y^2i+2x^3yj and c be the path consisting of line segments from(1,2) to (-1,3), from (-1,3) to (-1,1), and from (-1,1) to (2,1). evaluate the line integral of F along c.

Let F(x, y) = 3x²y2 i + 2x’yj and C be the path consisting of line segments from (1, 2) to (-1,3), from (-1, 3) to (-1, 1), and from (-1, 1) to (2, 1). Evaluate the line integral of F along C.

Answer 1

solm! Here, Flaiy) = 3x²y2? + 2x3y and c. be the path considering the path sting of line segmenta from (132) to t1,3), from (4,3) to tign and from £1,2] to (201) set a be the line segment from (1, 2) to (-133) ez be the line segment from +1, 3) to tlal). and C3 be the line segment from Holy to (2,1). Then c = 9.UCqY (3: Therefore, I F. dr as E. dr +SF. dr + $7.87 Let Phd ,y) = 3x²y2 and 8 xing) = 2x²y. Then of y = 62²y = do This shows that E is conservative. so, there is a scalar potential func- . tion f such that of a ... Now, df = of dn t of dy z ff. dr = F.dr. = 3x3y2 dx + 2x?ydy = dl. x3y2) on

Hence, on integration, we have - flasy) = x3y² + e, where cisan - arbitrary constant Therefore, {P* _F.47 [f.de +fF.de = 613) 6100 ,12,12 df +aft! If (1,27 1,3 ,1) = f(-1,3)-f(1/2) + f(-1,1) I ff1,3) +f(2,1) - {(-1919 = f(2,1) – f(1-2) = (8 +€) -(4+0)