Evaluate the following integral using any Vector Calculus method

Question

Evaluate the following integral using any Vector Calculus method
The integral with bounds C; [(4+e^yz)i+(xze^yz+2yz)j+(6z²+xye^yz+y²)k] dot dr, where C is the part of a curve on x⁵+y³z=25 from (1,2,3) to (2,-1,7);
Specific which method you used.

Answer 1

∫(4+e^yz) dx = 4x + xe^yz + f(y,z)

 

∫(xze^yz + 2yz) dy = xe^yz + y²z + g(x,z)

 

∫(6z² + xye^yz + y²) dz = 2z³ + xe^yz + y²z + h(x,y)

 

This is a conservative field since we can take

 

f(y,z) = y²z + 2z³

 

g(x,z) = 4x + 2z³

 

h(x,y) = 4x

 

Thus F(x,y,z) = ∇U(x,y,z) where

 

U(x,y,z) = 4x + y²z + 2z³ + xe^yz

 

Thus by the Fundamental theorem of line integrals, the answer is

 

U(2,-1,7) - U(1,2,3)

 

= (4·2 + (-1)²·7 + 2·7³ + 2e^(-1)·2) - (4·1 + 2²·3 + 2·3³ + 1·e^2·3 )

 

= (687 + 2e^-2) - (70 + e⁶)

 

= 617 - e⁶ + 2/e²