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Limits of Riemann sums as integrals practice problems

Recognize the limit of a Riemann sum as a definite integral, and evaluate it. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

2 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 4.1053

medium
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\left(1 + \frac{i - 1}{n}\right)^{2}}{n} \) as a definite integral, and evaluate it.
  1. Δx = 1/n, and x_i = 1 + 1(i − 1)/n runs across [1, 2] (left endpoints). The sum is Σ f(x_i)Δx with f(x) = x**2.
    Reviewed
  2. \[ \int\limits_{1}^{2} x^{2}\, dx = \frac{7}{3} \]
    Evaluate the integral.✓ Proved
  3. \[ \lim_{n \to \infty} \sum_{i=1}^{n} \frac{\left(1 + \frac{i - 1}{n}\right)^{2}}{n} = \frac{7}{3} \]
    The limit of the sum itself, from the summation formulas, agrees.✓ Proved
Answer \( \int_{1}^{2} x^{2}\, dx = \frac{7}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1449

medium
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{i^{3}}{n^{4}} \) as a definite integral, and evaluate it.
  1. Δx = 1/n, and x_i = 0 + 1i/n runs across [0, 1] (right endpoints). The sum is Σ f(x_i)Δx with f(x) = x**3.
    Reviewed
  2. \[ \int\limits_{0}^{1} x^{3}\, dx = \frac{1}{4} \]
    Evaluate the integral.✓ Proved
  3. \[ \lim_{n \to \infty} \sum_{i=1}^{n} \frac{i^{3}}{n^{4}} = \frac{1}{4} \]
    The limit of the sum itself, from the summation formulas, agrees.✓ Proved
Answer \( \int_{0}^{1} x^{3}\, dx = \frac{1}{4} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More limits of riemann sums as integrals practice

Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{18 i}{n} + 3}{n} \) as a definite integral, and evaluate it.
Problem 4.1047medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\left(\frac{i}{n} - 1\right)^{2}}{n} \) as a definite integral, and evaluate it.
Problem 4.1048medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{-4 + \frac{8 \left(4 i - 4\right)}{n}}{n} \) as a definite integral, and evaluate it.
Problem 4.1049medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\left(i - 1\right)^{3}}{n^{4}} \) as a definite integral, and evaluate it.
Problem 4.1050medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{16 i}{n} - 4}{n} \) as a definite integral, and evaluate it.
Problem 4.1051medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{4 \left(1 + \frac{4 i - 4}{n}\right)^{2}}{n} \) as a definite integral, and evaluate it.
Problem 4.1052medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{4 \left(4 i - 4\right)^{2}}{n^{3}} \) as a definite integral, and evaluate it.
Problem 4.1054medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{16 i^{3}}{n^{4}} \) as a definite integral, and evaluate it.
Problem 4.1055medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{2 \left(\frac{2 i}{n} + 2\right)^{2}}{n} \) as a definite integral, and evaluate it.
Problem 4.1056medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\left(\frac{i}{n} + 2\right)^{2}}{n} \) as a definite integral, and evaluate it.
Problem 4.1057medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 \left(-1 + \frac{i - 1}{n}\right)^{2} - 1}{n} \) as a definite integral, and evaluate it.
Problem 4.1058medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{2 \left(\frac{2 i}{n} - 1\right)^{3}}{n} \) as a definite integral, and evaluate it.
Problem 4.1059medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 + \frac{3 \left(3 i - 3\right)}{n}}{n} \) as a definite integral, and evaluate it.
Problem 4.1060medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\left(-1 + \frac{i - 1}{n}\right)^{2}}{n} \) as a definite integral, and evaluate it.
Problem 4.1061medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{3 \left(3 i - 3\right)}{n} + \frac{3 \left(3 i - 3\right)^{2}}{n^{2}}}{n} \) as a definite integral, and evaluate it.
Problem 4.1167medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{4 \left(\frac{4 i}{n} + 1\right)^{3}}{n} \) as a definite integral, and evaluate it.
Problem 4.1168medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{18 i}{n} + 9}{n} \) as a definite integral, and evaluate it.
Problem 4.1169medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 \left(1 + \frac{3 i - 3}{n}\right)^{2} + 3 + \frac{3 \left(3 i - 3\right)}{n}}{n} \) as a definite integral, and evaluate it.
Problem 4.1170medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{81 i^{2}}{n^{2}} - 3}{n} \) as a definite integral, and evaluate it.
Problem 4.1171medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{-3 + \frac{6 \left(3 i - 3\right)}{n}}{n} \) as a definite integral, and evaluate it.
Problem 4.1172medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 \left(2 + \frac{3 i - 3}{n}\right)^{2} + 6 + \frac{3 \left(3 i - 3\right)}{n}}{n} \) as a definite integral, and evaluate it.
Problem 4.1173medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{2 \left(2 i - 2\right)^{3}}{n^{4}} \) as a definite integral, and evaluate it.
Problem 4.1174medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{9 + \frac{6 \left(3 i - 3\right)}{n}}{n} \) as a definite integral, and evaluate it.
Problem 4.1175medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 \left(\frac{3 i}{n} - 1\right)^{2}}{n} \) as a definite integral, and evaluate it.
Problem 4.1176medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{4 \left(2 + \frac{4 i - 4}{n}\right)^{2}}{n} \) as a definite integral, and evaluate it.
Problem 4.1177medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\left(i - 1\right)^{2}}{n^{3}} \) as a definite integral, and evaluate it.
Problem 4.1178medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{64 i^{2}}{n^{3}} \) as a definite integral, and evaluate it.
Problem 4.1179medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 \left(\frac{3 i}{n} + 2\right)^{3}}{n} \) as a definite integral, and evaluate it.
Problem 4.1180medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{4 i}{n} + 2}{n} \) as a definite integral, and evaluate it.
Problem 4.1181medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{2 i}{n} - 1}{n} \) as a definite integral, and evaluate it.
Problem 4.1221medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{16 i}{n} + 4 \left(\frac{4 i}{n} + 1\right)^{2} + 4}{n} \) as a definite integral, and evaluate it.
Problem 4.1222medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{6 \left(\frac{2 i}{n} - 1\right)^{2} - 2}{n} \) as a definite integral, and evaluate it.
Problem 4.1223medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{6 \left(\frac{2 i}{n} + 1\right)^{2} - 2}{n} \) as a definite integral, and evaluate it.
Problem 4.1224medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{256 i^{3}}{n^{4}} \) as a definite integral, and evaluate it.
Problem 4.1225medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 \left(3 i - 3\right)^{3}}{n^{4}} \) as a definite integral, and evaluate it.
Problem 4.1226medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{-1 + \frac{2 \left(i - 1\right)}{n}}{n} \) as a definite integral, and evaluate it.
Problem 4.1227medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{16 i}{n} + 4}{n} \) as a definite integral, and evaluate it.
Problem 4.1228medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\left(2 + \frac{i - 1}{n}\right)^{3}}{n} \) as a definite integral, and evaluate it.
Problem 4.1229medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{-1 + \frac{3 \left(i - 1\right)^{2}}{n^{2}}}{n} \) as a definite integral, and evaluate it.
Problem 4.1230medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{6 \left(2 + \frac{2 i - 2}{n}\right)^{2} - 2}{n} \) as a definite integral, and evaluate it.
Problem 4.1288medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{9 \left(1 + \frac{3 i - 3}{n}\right)^{2} - 3}{n} \) as a definite integral, and evaluate it.
Problem 4.1289medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{2 \left(2 i - 2\right)}{n^{2}} \) as a definite integral, and evaluate it.
Problem 4.1290medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{12 \left(1 + \frac{4 i - 4}{n}\right)^{2} - 4}{n} \) as a definite integral, and evaluate it.
Problem 4.1291medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{2 \left(1 + \frac{2 i - 2}{n}\right)^{2}}{n} \) as a definite integral, and evaluate it.
Problem 4.1292medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{6 + \frac{3 \left(3 i - 3\right)}{n}}{n} \) as a definite integral, and evaluate it.
Problem 4.1293medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{4 i}{n^{2}} \) as a definite integral, and evaluate it.
Problem 4.1294medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{2 i}{n} + 3}{n} \) as a definite integral, and evaluate it.
Problem 4.1295medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 \left(\frac{3 i}{n} + 1\right)^{3}}{n} \) as a definite integral, and evaluate it.
Problem 4.1296medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\frac{16 i}{n} + 8}{n} \) as a definite integral, and evaluate it.
Problem 4.1297medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{9 i}{n^{2}} \) as a definite integral, and evaluate it.
Problem 4.1444medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 \left(\frac{i}{n} + 2\right)^{2} - 1}{n} \) as a definite integral, and evaluate it.
Problem 4.1445medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 + \frac{6 \left(3 i - 3\right)}{n}}{n} \) as a definite integral, and evaluate it.
Problem 4.1446medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\left(2 + \frac{i - 1}{n}\right)^{2} + 2 + \frac{i - 1}{n}}{n} \) as a definite integral, and evaluate it.
Problem 4.1447medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 \left(2 + \frac{3 i - 3}{n}\right)^{3}}{n} \) as a definite integral, and evaluate it.
Problem 4.1448medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{2 \left(2 + \frac{2 i - 2}{n}\right)^{3}}{n} \) as a definite integral, and evaluate it.
Problem 4.1450medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{4 \left(-1 + \frac{4 i - 4}{n}\right)^{3}}{n} \) as a definite integral, and evaluate it.
Problem 4.1451medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{4 \left(2 + \frac{4 i - 4}{n}\right)^{3}}{n} \) as a definite integral, and evaluate it.
Problem 4.1452medium✓ Every equation proved
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{-2 + \frac{6 \left(2 i - 2\right)^{2}}{n^{2}}}{n} \) as a definite integral, and evaluate it.
Problem 4.1453medium✓ Every equation proved