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To find the area of the shaded region between the curves \(y = 5 - x^2\) and \(y = x\), we will set up an integral representing the difference between the two functions over the interval where they intersect.

### Step 1: Find the Points of Intersection
To find the points where the two curves intersect, we set the equations equal to each other:

\[
5 - x^2 = x
\]

Rearranging the equation:

\[
x^2 + x - 5 = 0
\]

Solve this quadratic equation using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 1\), and \(c = -5\):

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-5)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 20}}{2} = \frac{-1 \pm \sqrt{21}}{2}
\]

The two points of intersection are:

\[
x_1 = \frac{-1 - \sqrt{21}}{2}, \quad x_2 = \frac{-1 + \sqrt{21}}{2}
\]

### Step 2: Set Up the Integral for the Area
The area between the curves from \(x_1\) to \(x_2\) is given by the integral:

\[
\text{Area} = \int_{x_1}^{x_2} \left( (5 - x^2) - x \right) \, dx
\]

Simplify the integrand:

\[
\text{Area} = \int_{x_1}^{x_2} (5 - x^2 - x) \, dx
\]

### Step 3: Compute the Integral
Now, integrate term by term:

\[
\int (5 - x^2 - x) \, dx = \int 5 \, dx - \int x^2 \, dx - \int x \, dx
\]

This results in:

\[
= 5x - \frac{x^3}{3} - \frac{x^2}{2} + C
\]

### Step 4: Evaluate the Definite Integral
Now, evaluate this expression from \(x_1 = \frac{-1 - \sqrt{21}}{2}\) to \(x_2 = \frac{-1 + \sqrt{21}}{2}\):

\[
\text{Area} = \left[5x - \frac{x^3}{3} - \frac{x^2}{2} \right]_{x_1}^{x_2}
\]

Substitute the limits of integration to find the area.

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Set up an integral for the area of the shaded region. Calculate the integral to find the area of the shaded region.

Learn how to find the area of the shaded region between the parabola y = 5 - x^2 and the line y = x using calculus. This detailed guide includes solving quadratic equations to find points of intersection, setting up an integral, and calculating it to determine the area.

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