📝 Riemann Sums And The Definite Integral

Q1. A dragster travels from \(t = 0\) to \(t = 5\) seconds with velocity \(v(t) = 4t + 1\) m/s. Using a left Riemann sum with 5 equal subintervals, what is the approximate distance traveled?

💡 Difficulty: easy | ✅ Correct: C

Q2. A car's speed is given by \(v(t) = 3t^2 + 2\) (m/s). Using Simpson's Rule with \(n = 4\) subintervals over \([0,2]\), estimate the distance traveled.

💡 Difficulty: medium | ✅ Correct: C

Q3. Prove that if \(f\) is continuous on \([a,b]\), then \(\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f\left( a + i \frac{b - a}{n} \right) \frac{b - a}{n}\).

💡 Difficulty: hard | ✅ Correct: D