Worksheet: Numerical Integration: Riemann Sums

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In this worksheet, we will practice using right, left, and midpoint Riemann sums to numerically approximate definite integrals.

Q1:

The table shows the values of a function obtained from an experiment. Estimate ο„Έ 𝑓 ( π‘₯ ) π‘₯    d using three equal subintervals with left endpoints.

π‘₯ 5 7 9 11 13 15 17
𝑓 ( π‘₯ ) βˆ’ 3 βˆ’ 1 . 7 βˆ’ 0 . 6 0.4 1.8 2.5 3.1

Q2:

Using the midpoint rule with 𝑛 = 5 , round ο„Έ 2 π‘₯ 3 π‘₯ + 2 π‘₯   d to four decimal places.

Q3:

The table shows the values of a function obtained from an experiment. Estimate ο„Έ 𝑓 ( π‘₯ ) π‘₯   d using three equal subintervals with midpoints.

π‘₯ 1 2 3 4 5 6 7
𝑓 ( π‘₯ ) βˆ’ 3 . 3 βˆ’ 2 . 1 βˆ’ 1 . 3 βˆ’ 0 . 1 0.9 2.1 3.1

Q4:

Estimate ο„Έ 5 ο€» 2 √ 3 π‘₯  π‘₯   s i n d using the midpoint rule with 𝑛 = 4 , giving your answer to four decimal places.

Q5:

Estimate ο„Έ βˆ’ 4 π‘₯ ( 5 π‘₯ ) π‘₯   οŽ„  s i n d using the midpoint rule with 𝑛 = 4 , giving your answer to four decimal places.

Q6:

Using the Midpoint Rule with 𝑛 = 5 , give an estimate of ο„Έ 5 √ 2 π‘₯ + 1 π‘₯    d . Give your answer to four decimal places.

Q7:

Approximate the integral ο„Έ ο€Ή 3 π‘₯ βˆ’ 5 π‘₯  π‘₯     d using a Riemann sum with right endpoints. Take 𝑛 to be 8.

  • A 6 3 4
  • B 4 3 2
  • C βˆ’ 6 3 4
  • D βˆ’ 2 3 2
  • E 2 3 2

Q8:

The table gives the values of a function obtained from an experiment. Use them to estimate ο„Έ 𝑓 ( π‘₯ ) π‘₯    d using three equal subintervals with right endpoints.

π‘₯ 3 7 11 15 19 23 27
𝑓 ( π‘₯ ) βˆ’ 2 . 6 βˆ’ 1 . 4 βˆ’ 0 . 7 0.8 2.3 3.4 4.8

Q9:

Calculate the midpoint rule estimate of ο„Έ π‘₯ + 2 π‘₯ οŠͺ   d with 𝑛 = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A16, an underestimate
  • B48, an overestimate
  • C16, an overestimate
  • D28, an underestimate
  • E28, an overestimate

Q10:

The table gives sampled values of an increasing function 𝑓 . Use the data to give a lower and upper bound for ο„Έ 𝑓 ( π‘₯ ) π‘₯     d .

π‘₯ 10 13 16 19 22 25
𝑓 ( π‘₯ ) βˆ’ 1 1 βˆ’ 3 1 4 8 10
  • Alower bound: βˆ’ 1 2 , upper bound: 66
  • Blower bound: βˆ’ 6 0 , upper bound: 3
  • Clower bound: βˆ’ 1 , upper bound: 20
  • Dlower bound: βˆ’ 2 0 , upper bound: 1
  • Elower bound: βˆ’ 3 , upper bound: 60

Q11:

Calculate the left endpoint estimate of ο„Έ π‘₯ + 2 π‘₯ οŠͺ   d with 𝑛 = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A16, an underestimate
  • B16, an overestimate
  • C48, an underestimate
  • D48, an overestimate
  • E18, an underestimate

Q12:

Calculate the right endpoint estimate of ο„Έ π‘₯ + 2 π‘₯ οŠͺ   d with 𝑛 = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A48, an underestimate
  • B13, an underestimate
  • C48, an overestimate
  • D16, an underestimate
  • E16, an overestimate

Q13:

Suppose that 𝐹 β€² ( π‘₯ ) = 3    and 𝐹 ( 0 ) = 7 . Find two estimates of 𝐹 ( 1 ) to three decimal places using the left and then the right endpoint methods with 8 rectangles.

  • A7.843, 7.686
  • B0.7689, 0.686
  • C7.769, 7.843
  • D7.686, 7.769
  • E7.769, 7.686

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