Riemann Sums

MATH 141 – Quiz 1 – More Sample Questions

Here’s a second set of sample questions from the first of this year’s 141 quizzes. As expected, the general structure appears consistent, but the relative level of difficulty is a bit higher here. For example, the Riemann Sum question doesn’t have lower limit a = 0 which means you really have to have a solid understanding of how to work backwards to identify the function. Thanks to Marco for sharing!

1. Evaluate \displaystyle \lim_{n\rightarrow \infty} \sum_{i = 1}^{n}\sin\left(\frac{\pi}{2} + \frac{\pi i}{2n} \right)\cdot \frac{\pi}{2n}

2. \displaystyle \frac{\text{d}}{\text{d}x} \int_{\sqrt{x}}^4 e^{t^2}\text{d}t =

3. Evaluate the following integrals:

    • \displaystyle \int \frac{\pi}{6+x^2} \text{d}x
    • \displaystyle \int \frac{x+5}{x^2 + 10x - 7}\text{d}x
    • \displaystyle \int_a^{b} \frac{1}{x(1+\ln^2 x)}\text{d}x