Problem 26 - Calculus Early Transcendentals

Question

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. $$\iint_{R} y \cos x y d A ; R=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq \pi / 3\\}$$

Step-by-Step Solution

Answer

The best order to evaluate the double integral is \(\int_0^{\pi/3} \int_0^1 y\cos(xy) dx \, dy\). The approximate value of the integral is \(0.7389\).

Step 1: Evaluate \(\int_0^1 \int_0^{\pi/3} y\cos(xy) dy \, dx\)

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.

Step 2: Evaluate \(\int_0^{\pi/3} \int_0^1 y\cos(xy) dx \, dy\)

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: $$\int_0^{\pi/3} \frac{\sin(y)}{y} dy$$

Step 3: Identify the Best Order of Integration and Evaluate

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 \(0.7389\).