Find the area in The region between the graphs of y = 4 − x2 and y = 2 − x from x = −2 to x = 2.

To find the area between the curves \( y = 4 - x^2 \) and \( y = 2 - x \) from \( x = -2 \) to \( x = 2 \), we proceed step by step:

### Step 1: Determine the points of intersection
First, we set the two functions equal to each other to find where they intersect:

\[
4 - x^2 = 2 - x
\]

Rearranging the equation:

\[
4 - 2 = x^2 - x
\]

\[
2 = x^2 - x
\]

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

Now, solve this quadratic equation using factoring:

\[
(x - 2)(x + 1) = 0
\]

Thus, the points of intersection are \( x = 2 \) and \( x = -1 \).

### Step 2: Set up the integral
To find the total area, we need to break the problem into two parts: one for the region between \( x = -2 \) and \( x = -1 \), and the other between \( x = -1 \) and \( x = 2 \).

1. For \( x = -2 \) to \( x = -1 \), the curve \( y = 4 - x^2 \) is above \( y = 2 - x \).
2. For \( x = -1 \) to \( x = 2 \), the curve \( y = 2 - x \) is above \( y = 4 - x^2 \).

The total area is the sum of the areas of these two regions:

\[
\text{Area} = \int_{-2}^{-1} \left[(4 - x^2) - (2 - x)\right] \, dx + \int_{-1}^{2} \left[(2 - x) - (4 - x^2)\right] \, dx
\]

### Step 3: Evaluate the integrals
1. First integral:
\[
\int_{-2}^{-1} \left[(4 - x^2) - (2 - x)\right] \, dx = \int_{-2}^{-1} (2 - x^2 + x) \, dx
\]

2. Second integral:
\[
\int_{-1}^{2} \left[(2 - x) - (4 - x^2)\right] \, dx = \int_{-1}^{2} (-2 + x^2 + x) \, dx
\]

Let me calculate the result.The total area between the curves \( y = 4 - x^2 \) and \( y = 2 - x \) from \( x = -2 \) to \( x = 2 \) is \( \frac{10}{3} \) square units.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you solve systems of equations by substitution to find intersection points?
2. What is the general formula for finding the area between two curves?
3. Can we use a different method, like the trapezoidal rule, to approximate the area?
4. How would the limits of integration change if the curves were symmetric about the y-axis?
5. What if we wanted to calculate the volume by rotating these curves around the x-axis?

**Tip:** Always check which curve is on top within the integration limits when calculating areas between curves!

Find the area in the region between the graphs of y = 4 − x^2 and y = 2 − x from x = −2 to x = 2.

Learn how to calculate the area between the curves y = 4 − x^2 and y = 2 − x from x = −2 to x = 2 using integral calculus. This step-by-step solution involves finding intersection points and setting up integrals to compute the exact area. Ideal for students studying Calculus I.

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