Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer.

Between y = x^2 − 4x + 1 and y = −x^2 + 4x − 5 for x in [0, 3]

Please don't skip any steps so I understand how to do the problem

We did this for x in [1,3]

Just change the limits of integration and follow the same steps. How is this one more difficult? Where do you get stuck?

To find the area of the region between the two curves, we need to follow a few steps. Let's break it down:

Step 1: Graph the curves
To start, let's graph the two curves to visualize the region we are interested in. You can use graphing software or an online graphing tool to plot the curves. Once you have the graphs, it will be easier to see the shape of the region.

Step 2: Determine the points of intersection
Look for the points where the two curves intersect. These are the x-values where the y-values of the two curves are equal. To find these points, set the two equations equal to each other and solve for x:

x^2 − 4x + 1 = −x^2 + 4x − 5

Simplify the equation:
2x^2 - 8x + 6 = 0

Factor or use the quadratic formula to solve for x. After finding the x-values of the points of intersection, make sure they lie within the interval [0, 3] that was given in the question.

Step 3: Determine which curve is above the other
Check the y-values of the two curves at the points of intersection found in step 2. Determine which curve is above the other for each segment of the interval [0, 3]. This information will help us set up the integral correctly.

Step 4: Set up the integral
To find the area of the region, we need to integrate the difference between the two curves. Since we already determined which curve is above the other for each segment of the interval, we can set up the integral accordingly:

∫[a, b] (top curve - bottom curve) dx

Here, a and b represent the x-values corresponding to the points of intersection and the interval [a, b] represents the limits of integration.

Step 5: Evaluate the integral
Use calculus or a numerical integration method (such as using technology) to evaluate the integral from step 4. This will give you the area of the region between the two curves.

Following these steps will allow you to find the area of the indicated region accurately.