Question

2. Consider the line integral I C F · d r, where the vector field F...

2. Consider the line integral I C F · d r, where the vector field F = x(cos(x 2 ) + y)i + 2y 3 (e y sin3 y + x 3/2 )j and C is the closed curve in the first quadrant consisting of the curve y = 1 − x 3 and the coordinate axes x = 0 and y = 0, taken anticlockwise.

(a) Use Green’s theorem to express the line integral in terms of a double integral over the region R enclosed by C.

(b) Calculate the double integral using a repeated integral with respect to y first and x second.

(c) Now reverse the order of integration and evaluate the double integral using a repeated integral with respect to x first and y second.

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