Answer :
The line integral over C alone is 60/3 - (-44) = 64.
What is integral?
Mathematicians define an integral as either a number equal to the area under the graph of a function for a certain interval or as a new function whose derivative is the original function.
solution:
It looks like the integral is
∫ xydx + x^2 y^3 dy
Let's close the loop by adding a line integral over the line segment joining (1, 4) to (0, 0). Then the closed loop is the triangular region
T = {(x, y) : 0 ≤ x ≤ 1 and 0 ≤ y ≤ 4x}
Since the integrand has no singularities over or on the boundary of T, we have by Green's theorem
Compute the double integral:
From this result, we subtract the line integral over the extra line segment we added. Parameterize this path by
C' : {(1 - t, 4 - 4t) : 0 ≤ t ≤ 1}
The line integral over C' is
∫ xydx + x^2 y^3 dy = ∫ ( 1 - t) ( 4 - 4t) ( -dt) + ( 1 - t)^2 ( 4 - 4t)^3 ( -4dt)
so that the line integral over C alone is 60/3 - (-44) = 64.
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