Integration

We need to handle the absolute value expressions carefully. Let’s analyze where each expression inside the absolute value changes sign:

• : changes at
• : changes at
• : changes at

So, we split the interval at :

1. Interval :

In this interval:

So the integrand becomes:

2. Interval :

In this interval:

So the integrand becomes:

Final Answer:

Area Bounded by the Curves

Given:

Find the area of the region bounded by the curves:

Step 1: Find points of intersection

Equate and : So the region of intersection lies between and , where:

Step 2: Set up the integral

Area between curves from to is: Here, from to :

Step 3: Evaluate the integral

✅ Final Answer:

We use partial fraction decomposition:

Multiply both sides by :

Expand both sides:

Now, equate coefficients:

Solving:

From

Substitute in

Substitute into third equation:

Now rewrite the integral:

Final Integration:

Final Answer: