Finding Area Bounded by Curves

Hi Ayliz, let's find the area bounded by the downward opening parabola, f of x equals six minus x squared, and the line, g of x equals x.

First, we need to find where these two graphs intersect. This will give us our limits of integration.

We set f of x equal to g of x. So, six minus x squared equals x.

Moving all terms to one side, we get x squared plus x minus six equals zero.

We can factor this quadratic equation into x plus three times x minus two equals zero.

Solving for x, we find our boundaries are x equals negative three and x equals two.

Now, let's visualize the region. Since f of x is a downward parabola and g of x is a line, the area is bounded between them. Over the interval from negative three to two, the parabola is above the line.

The area is the integral from negative three to two of the upper function minus the lower function.

Substituting our functions, we get the integral from negative three to two of six minus x squared minus x.

Now, we find the antiderivative of each term. Six becomes six x, negative x squared becomes negative x cubed over three, and negative x becomes negative x squared over two.