Calculate the area enclosed between the lines \( y = 2x + 3 \) and \( y = x^2 \).

To find the area enclosed between the lines \( y = 2x + 3 \) and \( y = x^2 \), we need to follow these steps: 1. **Find the points of intersection**: Set the equations equal to each other to find the x-values where they intersect. 2. **Determine the area between the curves**: Integrate the difference between the two functions from the left intersection point to the right intersection point. ### Step 1: Find the points of intersection Set the equations equal to each other: \[ 2x + 3 = x^2 \] Rearranging gives us: \[ x^2 - 2x - 3 = 0 \] Now, we can solve this quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = -2 \), and \( c = -3 \). Calculating the discriminant: \[ b^2 - 4ac = (-2)^2 - 4(1)(-3) = 4 + 12 = 16 \] Now substituting into the quadratic formula: \[ x = \frac{2 \pm \sqrt{16}}{2} = \frac{2 \pm 4}{2} \] This gives us two solutions: \[ x = \frac{6}{2} = 3 \quad \text{and} \quad x = \frac{-2}{2} = -1 \] So, the points of intersection are \( x = -1 \) and \( x = 3 \). ### Step 2: Determine the area between the curves The area \( A \) between the curves from \( x = -1 \) to \( x = 3 \) is given by: \[ A = \int_{-1}^{3} ((2x + 3) - (x^2)) \, dx \] Now, we can simplify the integrand: \[ A = \int_{-1}^{3} (2x + 3 - x^2) \, dx \] Next, we will calculate this integral. Let's perform the integration: \[ A = \int_{-1}^{3} (-x^2 + 2x + 3) \, dx \] Calculating the integral: \[ A = \left[ -\frac{x^3}{3} + x^2 + 3x \right]_{-1}^{3} \] Now, we will evaluate this expression at the bounds \( x = 3 \) and \( x = -1 \): 1. Evaluate at \( x = 3 \): \[ -\frac{3^3}{3} + 3^2 + 3(3) = -\frac{27}{3} + 9 + 9 = -9 + 9 + 9 = 9 \] 2. Evaluate at \( x = -1 \): \[ -\frac{(-1)^3}{3} + (-1)^2 + 3(-1) = -\frac{-1}{3} + 1 - 3 = \frac{1}{3} + 1 - 3 = \frac{1}{3} - 2 = \frac{1}{3} - \frac{6}{3} = -\frac{5}{3} \] Now, subtract the two results: \[ A = 9 - \left(-\frac{5}{3}\right) = 9 + \frac{5}{3} = \frac{27}{3} + \frac{5}{3} = \frac{32}{3} \] Thus, the area enclosed between the lines \( y = 2x + 3 \) and \( y = x^2 \) is \[ \boxed{\frac{32}{3}}. \]