Set up the definite integral required to find the area of the region between the graph of \( y=x^{2}-14 \) and \( y=4 x+7 \).

To find the area of the region between the graphs of \( y = x^2 - 14 \) and \( y = 4x + 7 \), we first need to determine the points of intersection of the two curves. We do this by setting the equations equal to each other: \[ x^2 - 14 = 4x + 7 \] Rearranging this equation gives: \[ x^2 - 4x - 21 = 0 \] Next, we can factor the quadratic: \[ (x - 7)(x + 3) = 0 \] This gives us the solutions: \[ x = 7 \quad \text{and} \quad x = -3 \] Now that we have the points of intersection, we can set up the definite integral to find the area between the curves. The area \( A \) between the curves from \( x = -3 \) to \( x = 7 \) is given by the integral of the top function minus the bottom function. To determine which function is on top, we can evaluate both functions at a point between the intersection points, say \( x = 0 \): \[ y = 0^2 - 14 = -14 \] \[ y = 4(0) + 7 = 7 \] Since \( 7 > -14 \), the line \( y = 4x + 7 \) is above the parabola \( y = x^2 - 14 \) in the interval \([-3, 7]\). Thus, the area \( A \) can be expressed as: \[ A = \int_{-3}^{7} \left( (4x + 7) - (x^2 - 14) \right) \, dx \] Simplifying the integrand: \[ A = \int_{-3}^{7} \left( 4x + 7 - x^2 + 14 \right) \, dx \] \[ = \int_{-3}^{7} \left( -x^2 + 4x + 21 \right) \, dx \] Therefore, the definite integral required to find the area of the region between the graphs is: \[ \int_{-3}^{7} (-x^2 + 4x + 21) \, dx \]