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.

  • A l i m 1 8 𝑛 3 + 6 𝑖 𝑛
  • B l i m 1 8 𝑛 3 + 6 𝑖 𝑛
  • C l i m 1 8 𝑛 3 6 𝑖 𝑛
  • D l i m 1 8 𝑛 3 + 6 𝑖 𝑛
  • E l i m 1 8 𝑛 6 𝑖 𝑛

Q2:

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

  • A l i m 7 𝑛 7 4 5 + 7 𝑖 𝑛
  • B l i m 7 𝑛 7 4 5 + 7 𝑖 𝑛
  • C l i m 7 𝑛 7 4 5 + 7 𝑖 𝑛
  • D l i m 7 𝑛 7 4 5 + 7 𝑖 𝑛
  • E l i m 7 𝑛 7 4 7 𝑖 𝑛

Q3:

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

  • A 𝑒 2 4 𝑥 𝑥 d
  • B 𝑒 2 4 𝑥 𝑥 d
  • C 𝑒 2 4 𝑥 𝑥 d
  • D 𝑒 2 4 𝑥 𝑥 d
  • E 𝑒 2 4 𝑥 𝑥 d

Q4:

Evaluate 𝑥3𝑥𝑥d using the limit of Riemann sums.

  • A38
  • B42
  • C16
  • D18
  • E36

Q5:

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

  • A 3 𝑒 5 𝑥 𝑥 d
  • B 3 𝑒 5 𝑥 𝑥 d
  • C 3 𝑒 5 𝑥 𝑥 d
  • D 3 𝑒 5 𝑥 𝑥 d
  • E 3 𝑒 5 𝑥 𝑥 d

Q6:

Express 5𝑥𝑥d as the limit of Riemann sums.

  • A l i m 1 0 𝑛 4 + 2 𝑖 𝑛
  • B l i m 1 0 𝑛 4 + 2 𝑖 𝑛
  • C l i m 1 0 𝑛 4 2 𝑖 𝑛
  • D l i m 1 0 𝑛 4 + 2 𝑖 𝑛
  • E l i m 1 0 𝑛 2 𝑖 𝑛

Q7:

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

  • A l i m 4 𝑛 3 1 + 4 𝑖 𝑛 + 2
  • B l i m 4 𝑛 3 1 + 4 𝑖 𝑛 + 2
  • C l i m 4 𝑛 3 1 + 4 𝑖 𝑛 + 2
  • D l i m 4 𝑛 3 1 + 4 𝑖 𝑛 + 2
  • E l i m 4 𝑛 3 4 𝑖 𝑛 + 2

Q8:

Evaluate 4𝑥4𝑥𝑥d by taking the limit of Riemann sums.

Q9:

Using Riemann sums, express lim𝑖𝑛 as an integral.

  • A 𝑥 𝑥 d
  • B 𝑥 𝑥 d
  • C 𝑥 𝑥 d
  • D 𝑥 𝑥 d
  • E 𝑥 𝑥 d

Q10:

Express 35𝑥𝑥sind as the limit of Riemann sums.

  • A l i m s i n 6 𝜋 𝑛 1 0 𝜋 𝑖 𝑛
  • B l i m s i n 6 𝜋 𝑛 1 0 𝜋 𝑖 𝑛
  • C l i m s i n 6 𝜋 𝑛 1 0 𝜋 𝑖 𝑛
  • D l i m s i n 6 𝜋 𝑛 1 0 𝜋 𝑖 𝑛
  • E l i m s i n 6 𝜋 𝑛 1 0 𝜋 𝑖 𝑛

Q11:

Evaluate 𝑥3𝑥5𝑥d using the limit of Riemann sums.

Q12:

Express lim1𝑛54 as a definite integral.

  • A 5 4 𝑥 𝑥 d
  • B 5 4 𝑥 𝑥 d
  • C 5 4 𝑥 𝑥 d
  • D 5 4 𝑥 𝑥 d
  • E 5 4 𝑥 𝑥 d

Q13:

Evaluate 𝑥2𝑥𝑥d using the limit of Riemann sums.

  • A 7 2 7
  • B 1 6
  • C 5 3
  • D 1 2
  • E 1 3

Q14:

Express 2𝑥5𝑥𝑥d as the limit of Riemann sums.

  • A l i m 3 𝑛 2 2 + 3 𝑖 𝑛 5 2 +
  • B l i m 3 𝑛 2 3 𝑖 𝑛 5 𝑛 3 𝑖
  • C l i m 3 𝑛 2 2 + 3 𝑖 𝑛 5 2 +
  • D l i m 3 𝑛 2 2 + 3 𝑖 𝑛 5 2 +
  • E l i m 3 𝑛 2 2 3 𝑖 𝑛 5 2

Q15:

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

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