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

To find the area under the curve defined by \( f(x) = 2x + 1 \) between \( x = 1 \) and \( x = 3 \), we'll use definite integration. Here's a step-by-step approach: ### 1. Set Up the Definite 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: \[ A = \int_{1}^{3} (2x + 1) \, dx \] ### 2. Find the Antiderivative First, find the indefinite integral (antiderivative) of \( f(x) \): \[ \int (2x + 1) \, dx = \int 2x \, dx + \int 1 \, dx = x^2 + x + C \] Here, \( C \) is the constant of integration, which cancels out in definite integrals. ### 3. Evaluate the Definite Integral Apply the limits of integration to the antiderivative: \[ A = \left[ x^2 + x \right]_{1}^{3} = (3^2 + 3) - (1^2 + 1) \] Calculate the values: \[ A = (9 + 3) - (1 + 1) = 12 - 2 = 10 \] ### 4. Conclusion The area under the curve \( f(x) = 2x + 1 \) from \( x = 1 \) to \( x = 3 \) is **10 square units**. --- **Visual Representation:** If you graph \( f(x) = 2x + 1 \) between \( x = 1 \) and \( x = 3 \), the area under the curve forms a trapezoid. The calculation above confirms that the area of this trapezoid is 10.