Direct integrate please

Question

Evaluate the integral

∲(e^x+y²)dx+(e^y+x²)dy

by evaluating it directly( without using green's theorem)

Assume that c is the boundary of the region between y=x² and y= x which is oriented counterclockwise.

Please help me in integrating directly

Answer 1

INtersection: x² = x; x = 0 or x = 1

We start from point (0, 0) and moving by parabola and then by strait line::

If y = x² then dy = 2xdx; if y = x then dy = dx

∲(e^x+y²)dx+(e^y+x²)dy = ∫₀¹[(e^x + x⁴) + (e^{x^2}+ x²)·2x]dx + ∫₁⁰(e^x + x² + e^x + x²)dx =

(e^x + x⁵/5 + e^{x^2} + x⁴/2)₀¹ + (2e^x + 2x³/3)₁⁰ = 2e + 1/5 + 1/2 - 2 + 2 - 2e - 2/3 = 1/30