Farmer jones, and his wife, dr. jones, decide to build a fence in their field, to keep the sheep safe. since dr. jones is a mathematician, she suggests building fences described by y=6x2 and y=x2+10. farmer jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to his wife. what is the area of the enclosed region? 42.164

Consider the attached figure (note that the axis are not in a 1:1 scale).

We want the (finite) area between the two parabolas, where is the green one, and is the red one.

We need the following result: given a function f, the integral

returns the area between the function and the x axis.

So, if we subtract the area below the green function between points A and B from the area below the red function between points A and B, we have exactly the area between the two. So, we need the x coordinates of points A and B.

Since they are the points of interception between the two parabolas, we can compute them by solving the equation

So, we need

The primitive of this function is

And we need to compute it at the two endpoints: the area is given by

it would open into a square depending on the type of envelope i’m not sure about the second part