vector integration

Question

Question- Evaluate by Green's theorem for $$\oint_C\frac{1}{y} dx + \frac{1}{x}dy$$where C is boundary of the region defined by $x=1, x=4, y=1, y^2=x.$ I solved this problem and got $-27/4$ as the answer but it seems to be wrong. I'd greatly appreciate some help in solving this problem.

Answer 1

Green theorem tells us:

$$\oint_{C^{+}} P(x,y)dx + Q(x,y)dy = \iint_G\left(\frac{\delta Q}{\delta x} - \frac{\delta P}{\delta y}\right)dO $$

Clearly, here, $P(x,y) = 1/y, Q(x,y) = 1/x$

Now $$\frac{\delta P}{\delta y} = -\frac{1}{y^2}, \frac{\delta Q}{\delta x} = -\frac{1}{x^2}$$

Hence, it suffices to calculate:

$$\iint_G\left(\frac{\delta Q}{\delta x} - \frac{\delta P}{\delta y}\right)dO = \iint_G\left(-1/x^2 + 1/y^2\right)dO$$

$$= \int\limits_1^4dx\int\limits_{1}^{\sqrt{x}}(-1/x^2 + 1/y^2)dy$$

$$= \int\limits_1^4dx[-y/x^2 - 1/y]_1^{\sqrt{x}} $$

$$= \int\limits_1^4dx(-x^{-3/2}-x^{-1/2} + x^{-2} + 1)$$

$$= \frac{3}{4}$$