91Ó°ÊÓ

First, let's consider the first order of integration: $$\int_0^1 \int_0^{\pi/3} y\cos(xy) dy \, dx$$ To find the inner integral, we integrate with respect to y: $$\int_0^1 \left[ \frac{\sin(xy)}{x} \Big|_0^{\pi/3} \right] dx$$ Next, we substitute the limits of integration for y: $$\int_0^1 \left[ \frac{\sin(x\pi/3)}{x} - \frac{\sin(0)}{x} \right] dx$$ Since \(\sin(0) = 0\), the integral becomes: $$\int_0^1 \frac{\sin(x\pi/3)}{x} dx$$ Now, we will evaluate the second order of integration.

We consider the second order of integration: $$\int_0^{\pi/3} \int_0^1 y\cos(xy) dx \, dy$$ To find the inner integral, we integrate with respect to x: $$\int_0^{\pi/3} \left[ \frac{\sin(xy)}{y} \Big|_0^1 \right] dy$$ Next, we substitute the limits of integration for x: $$\int_0^{\pi/3} \left[ \frac{\sin(y)}{y} - \frac{\sin(0)}{y} \right] dy$$ Again, since \(\sin(0) = 0\), the integral becomes: $$\int_0^{\pi/3} \frac{\sin(y)}{y} dy$$

Comparing both orders, we notice that \(\int_0^{\pi/3} \int_0^1 y\cos(xy) dx \, dy\) has a simpler expression. In this case, the second order of integration is the best one to evaluate the integral. To find the value of the integral, we compute: $$\int_0^{\pi/3} \frac{\sin(y)}{y} dy$$ Unfortunately, there is no elementary function for the antiderivative of \( \frac{\sin(y)}{y}\), so we can not find an exact value. But we can evaluate it numerically to obtain an approximate value. Using a numerical integration method, like Simpson's rule or a computer software, we find the approximate value of the integral to be: $$\int_0^{\pi/3} \frac{\sin(y)}{y} dy \approx 0.7389$$ Therefore, the best order to evaluate the double integral is \(\int_0^{\pi/3} \int_0^1 y\cos(xy) dx \, dy\) and the approximate value of the integral is \(0.7389\).