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Lesson: Applying Midpoint Approximation to Find the Area under a Curve

Worksheet • 14 Questions

Q1:

Let 𝑓 ( π‘₯ ) = 5 4 π‘₯ over 1 ≀ π‘₯ ≀ 2 . Using four subintervals and taking midpoints as sample points, evaluate the Riemann sum of 𝑓 to six decimal places.

Q2:

Let 𝑓 ( π‘₯ ) = 1 5 π‘₯ over 1 ≀ π‘₯ ≀ 5 . Using four subintervals and taking midpoints as sample points, evaluate the Riemann sum of 𝑓 to six decimal places.

Q3:

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

Q4:

Using the midpoint rule with 𝑛 = 5 , round ο„Έ 3 π‘₯ 2 π‘₯ βˆ’ 5 π‘₯ 4 3 d to four decimal places.

Q5:

The table shows the values of a function obtained from an experiment. Estimate ο„Έ 𝑓 ( π‘₯ ) π‘₯ 7 1 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

Q6:

The table shows the values of a function obtained from an experiment. Estimate ο„Έ 𝑓 ( π‘₯ ) π‘₯ 1 9 1 d using three equal subintervals with midpoints.

π‘₯ 1 4 7 10 13 16 19
𝑓 ( π‘₯ ) βˆ’ 3 . 6 βˆ’ 2 . 1 βˆ’ 1 . 2 βˆ’ 0 . 5 0.4 1.2 2

Q7:

The table shows the values of a function obtained from an experiment. Estimate ο„Έ 𝑓 ( π‘₯ ) π‘₯ 2 5 1 d using three equal subintervals with midpoints.

π‘₯ 1 5 9 13 17 21 25
𝑓 ( π‘₯ ) βˆ’ 4 βˆ’ 2 . 7 βˆ’ 2 βˆ’ 0 . 8 0.2 0.9 1.5

Q8:

The table shows the values of a function obtained from an experiment. Estimate ο„Έ 𝑓 ( π‘₯ ) π‘₯ 1 0 4 d using three equal subintervals with midpoints.

π‘₯ 4 5 6 7 8 9 10
𝑓 ( π‘₯ ) βˆ’ 2 . 9 βˆ’ 2 . 4 βˆ’ 1 . 9 βˆ’ 1 . 1 0.1 1.1 1.7

Q9:

Estimate ο„Έ 5 ο€» 2 √ 3 π‘₯  π‘₯ 9 1 s i n d using the midpoint rule with 𝑛 = 4 , giving your answer to four decimal places.

Q10:

Estimate using the midpoint rule with , giving your answer to four decimal places.

Q11:

Estimate using the midpoint rule with , giving your answer to four decimal places.

Q12:

Using the Midpoint Rule with 𝑛 = 5 , give an estimate of ο„Έ 5 √ 2 π‘₯ + 1 π‘₯ 1 0 3 d . Give your answer to four decimal places.

Q13:

Given 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 2 and βˆ’ 4 ≀ π‘₯ ≀ 2 , evaluate the Riemann sum for 𝑓 with six subintervals, taking sample points to be midpoints.

  • A βˆ’ 1 2
  • B5
  • C 1 2
  • D βˆ’ 5
  • E7

Q14:

Given 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 4 2 and βˆ’ 1 ≀ π‘₯ ≀ 5 , evaluate the Riemann sum for 𝑓 with six subintervals, taking sample points to be midpoints.

  • A59
  • B βˆ’ 8 6
  • C βˆ’ 5 9
  • D86
  • E38
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