**Problem Statement:**Find a limit equal to \[\int_{3}^{7} (x^2 - 8) \, dx\]**Explanation:**The integral given is a definite integral, which calculates the area under the curve of the function \(f(x) = x^2 - 8\) from \(x = 3\) to \(x = 7\). To evaluate this integral, we should find the antiderivative of the function \(f(x)\) and then apply the Fundamental Theorem of Calculus.1. **Antiderivative:** The function \(f(x) = x^2 - 8\) can be integrated to find its antiderivative: \[ \int (x^2 - 8) \, dx = \frac{x^3}{3} - 8x + C \]2. **Evaluate from 3 to 7:** Using the limits of integration, we calculate: \[ \left[ \frac{x^3}{3} - 8x \right]_{3}^{7} = \left( \frac{7^3}{3} - 8 \times 7 \right) - \left( \frac{3^3}{3} - 8 \times 3 \right) \]3. **Solve:** Calculate each part separately: - For \(x = 7\): \[ \frac{7^3}{3} - 8 \times 7 = \frac{343}{3} - 56 \] - For \(x = 3\): \[ \frac{3^3}{3} - 8 \times 3 = \frac{27}{3} - 24 \] Subtract the second result from the first to find the value of the definite integral.This process will yield the limit that is equal to the value of the definite integral.

The integral of some functions are given below. 1)∫x2dx=x33+c2)∫8dx=8x+c Use the above and find the value of the integral. ∫37(x2-8)dx=x33-8x37Apply the limits and simplify the integral further implies, x33-8x37=733-(8×7)-333+(8×3)=3433-56-9+24=3433-41=2203 Therefore, the answer is 2203.