Worksheet: Definite Integrals as Limits of Riemann Sums

In this worksheet, we will practice interpreting a definite integral as the limit of a Riemann sum when the size of the partitions tends to zero.

Q1:

Express 3𝑥𝑥d as the limit of Riemann sums.

  • Alim18𝑛3+6𝑖𝑛
  • Blim18𝑛3+6𝑖𝑛
  • Clim18𝑛36𝑖𝑛
  • Dlim18𝑛3+6𝑖𝑛
  • Elim18𝑛6𝑖𝑛

Q2:

Without evaluating the limit, express 74𝑥𝑥d as a limit of Riemann sums.

  • Alim7𝑛745+7𝑖𝑛
  • Blim7𝑛745+7𝑖𝑛
  • Clim7𝑛745+7𝑖𝑛
  • Dlim7𝑛745+7𝑖𝑛
  • Elim7𝑛747𝑖𝑛

Q3:

Express lim𝑒24𝑥Δ𝑥 as a definite integral on the interval [5,3].

  • A𝑒24𝑥𝑥d
  • B𝑒24𝑥𝑥d
  • C𝑒24𝑥𝑥d
  • D𝑒24𝑥𝑥d
  • E𝑒24𝑥𝑥d

Q4:

Evaluate 𝑥3𝑥𝑥d using the limit of Riemann sums.

Q5:

Express lim3𝑒5𝑥Δ𝑥 as a definite integral on the interval [5,1].

  • A3𝑒5𝑥𝑥d
  • B3𝑒5𝑥𝑥d
  • C3𝑒5𝑥𝑥d
  • D3𝑒5𝑥𝑥d
  • E3𝑒5𝑥𝑥d

Q6:

Express 5𝑥𝑥d as the limit of Riemann sums.

  • Alim10𝑛4+2𝑖𝑛
  • Blim10𝑛4+2𝑖𝑛
  • Clim10𝑛42𝑖𝑛
  • Dlim10𝑛4+2𝑖𝑛
  • Elim10𝑛2𝑖𝑛

Q7:

Without evaluating the limit, express 3𝑥+2𝑥d as a limit of Riemann sums.

  • Alim4𝑛31+4𝑖𝑛+2
  • Blim4𝑛31+4𝑖𝑛+2
  • Clim4𝑛31+4𝑖𝑛+2
  • Dlim4𝑛31+4𝑖𝑛+2
  • Elim4𝑛34𝑖𝑛+2

Q8:

Evaluate 4𝑥4𝑥𝑥d by taking the limit of Riemann sums.

Q9:

Using Riemann sums, express lim𝑖𝑛 as an integral.

  • A𝑥𝑥d
  • B𝑥𝑥d
  • C𝑥𝑥d
  • D𝑥𝑥d
  • E𝑥𝑥d

Q10:

Express 35𝑥𝑥sind as the limit of Riemann sums.

  • Alimsin6𝜋𝑛10𝜋𝑖𝑛
  • Blimsin6𝜋𝑛10𝜋𝑖𝑛
  • Climsin6𝜋𝑛10𝜋𝑖𝑛
  • Dlimsin6𝜋𝑛10𝜋𝑖𝑛
  • Elimsin6𝜋𝑛10𝜋𝑖𝑛

Q11:

Evaluate 𝑥3𝑥5𝑥d using the limit of Riemann sums.

Q12:

Express lim1𝑛54 as a definite integral.

  • A54𝑥𝑥d
  • B54𝑥𝑥d
  • C54𝑥𝑥d
  • D54𝑥𝑥d
  • E54𝑥𝑥d

Q13:

Evaluate 𝑥2𝑥𝑥d using the limit of Riemann sums.

  • A727
  • B16
  • C53
  • D12
  • E13

Q14:

Express 2𝑥5𝑥𝑥d as the limit of Riemann sums.

  • Alim3𝑛22+3𝑖𝑛52+
  • Blim3𝑛23𝑖𝑛5𝑛3𝑖
  • Clim3𝑛22+3𝑖𝑛52+
  • Dlim3𝑛22+3𝑖𝑛52+
  • Elim3𝑛223𝑖𝑛52

Q15:

Evaluate (𝑥4)𝑥d using the limit of Riemann sums.

Q16:

Evaluate 𝑥2𝑥+3𝑥d using the limit of Riemann sums.

  • A283
  • B23
  • C163
  • D203
  • E323

Q17:

Evaluate 𝑥𝑥d using the limit of Riemann sums.

  • A32
  • B20
  • C16
  • D20
  • E26

Q18:

Evaluate (𝑥2)𝑥d using the limit of Riemann sums.

  • A4
  • B8
  • C10
  • D7
  • E6

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.