Question

The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{4} \int_{\sqrt{x}}^{2} \frac{x}{y^{5}+1} d y d x$$

Short answer

#tag_title#Step 2: Reverse the order of integration #tag_content#To reverse the order of integration, we need to find the new limits for \(x\) and \(y\). For \(y\), the limits are still given by the line \(y=2\) and the curve \(y=\sqrt{x}\), but they will be expressed differently. The curve \(y=\sqrt{x}\) can be rewritten as \(x=y^2\). Thus, the new limits for \(y\) are \(0 \leq y \leq 2\). For \(x\), observe how it ranges between the curve \(x=y^2\) and the line \(x=4\) for each fixed value of \(y\). Therefore, the new limits for \(x\) are \(y^2 \leq x \leq 4\). Now, the order of integration has been reversed, and the new integral is written as \(\int_{0}^{2}\int_{y^2}^{4} f(x, y) \, dx\, dy\). #tag_title#Step 3: Evaluate the integral #tag_content#Now that we have reversed the order of integration, we can proceed to evaluate the integral. First, integrate with respect to \(x\), treating \(y\) as a constant. Then, integrate with respect to \(y\). Once both integrations are completed, the result will be the final value of the double integral over the given region.

Step-by-step solution

Sketch the region of integration

The given region of integration is determined by the limits involving \(x\) and \(y\): \(0 \leq x \leq 4\) and \(\sqrt{x} \leq y \leq 2\). To sketch these limits, first, draw the curve \(y = \sqrt{x}\), the line \(y=2\), and the vertical lines \(x=0\) and \(x=4\). The region of integration is enclosed by these curves and lines.