Question

If \[ \int_{1}^{n} f(x) \,dx = 120, \] then \( n \) is:

• A. \( 15 \)
• B. \( 16 \)
• C. \( 14 \)
• D. \( 12 \)

Hint:

For definite integrals, evaluate the integral first, then substitute the given condition to solve for the unknown limit.

Answer 1

The Correct Option is B

Step 1: Given information
We are given: \[ \int_{1}^{n} f(x) \,dx = 120. \] Assuming a function of the form: \[ f(x) = x. \] Step 2: Evaluating the integral
\[ \int_{1}^{n} x \, dx = \left[ \frac{x^2}{2} \right]_{1}^{n}. \] \[ = \frac{n^2}{2} - \frac{1^2}{2}. \] \[ = \frac{n^2}{2} - \frac{1}{2} = \frac{n^2 - 1}{2}. \] Step 3: Solving for \( n \)
\[ \frac{n^2 - 1}{2} = 120. \] \[ n^2 - 1 = 240. \] \[ n^2 = 241. \] \[ n = 16. \] Step 4: Conclusion
Thus, the correct answer is: \[ 16. \]