Question

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

Short answer

The reversed order of integration for the given double integral is: $$ \int_{0}^{4} \int_{\frac{y}{2}}^{\sqrt{y}} f(x, y) d x d y $$

Step-by-step solution

Identify the original region D

The region D in the xy-plane can be described by the given limits of integration: $$ 0 \le x \le 2 \\ x^2 \le y \le 2x $$ This gives us a region in the xy-plane bounded by the parabola \(y=x^2\), the straight line \(y=2x\), and the line \(x=0\).

Find the new limits of integration for switching the order

In order to reverse the order of integration, we need to rewrite the inequalities such that x limits involve y, and y limits involve x. So, we need to find the range of y and the range of x for given y. The minimum value of y occurs when \(x=0\), and y is also \(0\); the maximum value of y occurs when the equations \(y=x^2\) and \(y=2x\) intersect. So, let's find that point first: $$ x^2 = 2x \\ x^2 - 2x = 0 \\ x(x-2)=0 $$ The values of x at the intersection are \(x=0\) and \(x=2\). Since we already considered \(x=0\), we will consider the point when \(x=2\), and at this point, \(y=2\times2=4\). So, the y limits are: $$ 0 \le y \le 4 $$ Now, let's find the range of x for a given y. We'll express x in terms of y for the given two curves: $$ x=\sqrt{y} \quad (\text{for the parabola } y=x^2) \\ x=\frac{y}{2} \quad (\text{for the line } y=2x) $$ So, the x limits for a given y are: $$ \frac{y}{2} \le x \le \sqrt{y} $$

Rewrite the double integral with the new order of integration

Now we have the new limits of integration, we can rewrite the given double integral in terms of dy dx: $$ \int_{0}^{4} \int_{\frac{y}{2}}^{\sqrt{y}} f(x, y) d x d y $$ So, the reversed order of integration for the given double integral is: $$ \int_{0}^{4} \int_{\frac{y}{2}}^{\sqrt{y}} f(x, y) d x d y $$

Key concepts

Order of Integration

When working with double integrals, one of the crucial decisions is the order of integration. In a double integral, like the one given in the exercise, you are initially given an order, for example, first integrating with respect to \(y\) and then \(x\), represented as \(\int_0^2 \int_{x^2}^{2x} f(x, y) \, dy \, dx\). Reverse the order means changing it to \(\int \int \, dx \, dy\) or vice versa. In this exercise, we need to switch from integrating first with respect to \(y\) and then \(x\), to doing \(x\) first, followed by \(y\). This involves recalculating the integration bounds. Reversing the order can simplify calculations, depending on the function being integrated.

Bounds of Integration

The bounds of integration define the region over which we integrate. They are determined by the limits given in the integral and represent the area in the xy-plane where the function is being analyzed. In the given problem, the bounds are initially \(0 \leq x \leq 2\) and \( x^2 \leq y \leq 2x\). These determine the shape and limits of the region D. Visually, this represents a region bounded by the parabola \(y = x^2\) and the line \(y = 2x\). To adjust these bounds for a different order of integration, we look for intersections and create new y-limits, here calculated as \(0 \leq y \leq 4\). For each \(y\), \(x\) ranges from \(x = \frac{y}{2}\) to \(x = \sqrt{y}\). This new set of limits allows us to integrate in a new order effectively.

Integration Techniques

In the context of double integrals, choosing the right technique is key for simplification. Reversing the order of integration is often used when the original order results in complex or unsolvable integrals. By reevaluating the integrals under new bounds, we might access a more straightforward path to a solution. Given \(\int_{0}^{4} \int_{\frac{y}{2}}^{\sqrt{y}} f(x, y) \, dx \, dy\), we clearly identify that for each fixed \(y\), \(x\) varies from \(\frac{y}{2}\) to \(\sqrt{y}\), simplifying our work. Understanding the geometric meaning of the regions helps tailor our technique choice further. Practice utilizing different orders and comparing ease of calculations fosters an adeptness that benefits handling more challenging integrals in the future.