Question

Reverse the order of integration in the following integrals. $$\int_{0}^{3} \int_{0}^{6-2 x} f(x, y) d y d x$$

Short answer

Question: Reverse the order of integration for the given double integral ∫∫(0 to 3 for x and 0 to 6 - 2x for y) f(x, y) dy dx, and provide a graphical understanding of the problem. Answer: When the order of integration is reversed, the double integral becomes ∫∫(0 to 6 for y and 0 to (6-y)/2 for x) f(x, y) dx dy. The region of integration is a triangle with vertices at (0,0), (3,0), and (3,6).

Step-by-step solution

Identify the bounds of integration

First, we'll identify the bounds of x and y from the given integral: For x: 0 to 3 For y: 0 to 6 - 2x We use these bounds to draw the region and identify the new bounds of integration when reversing the order.

Draw the region

To help you visualize the region, plot the following lines: 1. x = 0 (vertical line along the y-axis) 2. x = 3 (vertical line parallel to the y-axis at x = 3) 3. y = 6 - 2x (line with a negative slope) When you plot these lines, you'll see that the region is a traingle. The x-axis spans from 0 to 3 while the y-axis spans from 0 to 6.

Reverse the order of integration

Now, we want to reverse the order of integration, which means we'll do dx dy instead of dy dx. To do this, we need to find new limits for x and y in terms of each other. Observe the plot, and you'll notice y spans from 0 to the line y = 6 - 2x. Rewrite this equation in terms of x: $$x = \frac{6 - y}{2}$$ Since we are reversing the order of integration, the outer integral now corresponds to y, which spans from 0 to 6 (as observed from the plot). The inner integral corresponds to x, which spans from x=0 to x=(6-y)/2. Therefore, the reversed order of integration is: $$\int_{0}^{6} \int_{0}^{\frac{6 - y}{2}} f(x, y) d x d y$$