Evaluate(2,1)∫(0,0)(10x4 − 2xy3)dx − (3x2y2)dy along the path x4 − 6xy3 = 4y2

To evaluate the given expression, we need to find the integral of the given function over the specified path.

Step 1: Find the equation of the path The given path is defined by the equation x^4 - 6xy^3 = 4y^2.

Step 2: Solve the equation for x Rearrange the equation to solve for x: x^4 - 6xy^3 = 4y^2 x^4 - 6xy^3 - 4y^2 = 0

Step 3: Substitute the value of x in terms of y into the integral Now that we have the equation of the path in terms of y, we can substitute the value of x in terms of y into the integral expression: ∫(0,0)(10x^4 - 2xy^3)dx - (3x^2y^2)dy

Step 4: Evaluate the integral Evaluate the integral with respect to x, treating y as a constant: ∫(0,0)(10x^4 - 2xy^3)dx = [2x^5 - xy^3] evaluated from x = 0 to x = 0 = 2(0)^5 - 0(0)^3 - [2(0)^5 - 0(0)^3] = 0

Step 5: Simplify the expression Now we have: 0 - (3x^2y^2)dy

Step 6: Evaluate the integral with respect to y Evaluate the integral with respect to y: ∫(0,0)(0 - 3x^2y^2)dy = -3x^2(1/3)y^3 evaluated from y = 0 to y = 0 = -3x^2(1/3)(0)^3 - (-3x^2(1/3)(0)^3) = 0

Step 7: Final result The final result of the integral is 0.