Question

In Exercises \(37-42,\) sketch the region \(R\) of integration and switch the order of integration. $$ \int_{-1}^{1} \int_{x^{2}}^{1} f(x, y) d y d x $$

Short answer

The region of integration is the area bounded by the parabola \(y=x^{2}\) and the line \(y=1\), with \(x\) ranging from -1 to 1. After changing the order of integration, the new integral is: \[\int_{0}^{1} \int_{-\sqrt{y}}^{\sqrt{y}} f(x, y) d x d y \]The short answer to this problem is the modified double integral.

Step-by-step solution

Identify the region of integration

The region of integration can be identified by the limits of the integral. For this integral, you can find that x ranges from -1 to 1, and within that, y varies from \(x^{2}\) to 1. This represents a region in the x-y plane bounded by the parabola \(y = x^{2}\) and ground \(y=1\), moving along x from -1 to 1.

Sketch the region of integration

After determining the limits of y and x, plot the region of integration, which represents the area bounded by \(y = x^{2}\) and \(y = 1\), with \(x\) ranging from -1 to 1.

Change the order of integration

To change the order of integration, remove dy dx and replace it with dx dy, which changes the role of x and y. Observe the region carefully to set the new limits. Here x will have definite limits, \(x\) ranges from 0 to 1, and for a particular x, y ranges from -sqrt(x) to sqrt(x)

Write down the new integral

After changing the limit, you have a new integral which is: $$ \int_{0}^{1} \int_{-\sqrt{y}}^{\sqrt{y}} f(x, y) d x d y $$