Find the area under the curve defined by $$ f(x) = 2x + 1 $$ between the points $$ x = 1 $$ and $$ x = 3 $$.

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To find the area under the curve defined by $$ f(x) = 2x + 1 $$ between $$ x = 1 $$ and $$ x = 3 $$, we can use definite integration. Here's a step-by-step guide to solving the problem:

### Step 1: Set Up the Integral
The area $$ A $$ under the curve from $$ x = a $$ to $$ x = b $$ is given by:

$$
A = \int_{a}^{b} f(x) \, dx
$$

For this problem, $$ f(x) = 2x + 1 $$, $$ a = 1 $$, and $$ b = 3 $$. Therefore:

$$
A = \int_{1}^{3} (2x + 1) \, dx
$$

### Step 2: Find the Antiderivative
To integrate $$ 2x + 1 $$, find the antiderivative of each term separately.

$$
\int (2x) \, dx = x^2 \quad \text{and} \quad \int 1 \, dx = x
$$

So, the antiderivative of $$ 2x + 1 $$ is:

$$
F(x) = x^2 + x
$$

### Step 3: Evaluate the Antiderivative at the Bounds
Apply the Fundamental Theorem of Calculus by evaluating $$ F(x) $$ at $$ x = 3 $$ and $$ x = 1 $$:

$$
A = F(3) - F(1) = (3^2 + 3) - (1^2 + 1)
$$

Calculate each part:

$$
F(3) = 9 + 3 = 12
$$
$$
F(1) = 1 + 1 = 2
$$

### Step 4: Subtract to Find the Area
$$
A = 12 - 2 = 10
$$

### Conclusion
The area under the curve $$ f(x) = 2x + 1 $$ from $$ x = 1 $$ to $$ x = 3 $$ is **10 square units**.
The area under the curve $$ f(x) = 2x + 1 $$ between $$ x = 1 $$ and $$ x = 3 $$ is **10 square units**.

Find the area under the curve defined by \( f(x) = 2x + 1 \) between the points \( x = 1 \) and \( x = 3 \).

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