Worksheet: Finding Areas Using Midpoint Riemann Sum

In this worksheet, we will practice finding an approximated value of the area under a curve using midpoint approximation.

Q1:

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

Q2:

The table shows the values of a function obtained from an experiment. Estimate 𝑓 ( 𝑥 ) 𝑥 d using three equal subintervals with midpoints.

𝑥 1 2 3 4 5 6 7
𝑓 ( 𝑥 ) 3 . 3 2 . 1 1 . 3 0 . 1 0.9 2.1 3.1

Q3:

Given 𝑓 ( 𝑥 ) = 𝑥 4 2 and 4 𝑥 2 , evaluate the Riemann sum for 𝑓 with six subintervals, taking sample points to be midpoints.

  • A7
  • B 1 2
  • C 5
  • D 1 2
  • E5

Q4:

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

Q5:

Using the Midpoint Rule with 𝑛 = 5 , give an estimate of 5 2 𝑥 + 1 𝑥 1 0 3 d . Give your answer to four decimal places.

Q6:

Using the midpoint rule with 𝑛 = 5 , round 2 𝑥 3 𝑥 + 2 𝑥 5 2 d to four decimal places.

Q7:

Estimate 5 2 3 𝑥 𝑥 s i n d using the midpoint rule with 𝑛 = 4 , giving your answer to four decimal places.

Q8:

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

Q9:

Using the midpoint rule with 𝑛 = 5 , round 3 𝑥 2 𝑥 5 𝑥 4 3 d to four decimal places.

Q10:

The table shows the values of a function obtained from an experiment. Estimate 𝑓 ( 𝑥 ) 𝑥 d using three equal subintervals with midpoints.

𝑥 1 4 7 10 13 16 19
𝑓 ( 𝑥 ) 3 . 6 2 . 1 1 . 2 0 . 5 0.4 1.2 2

Q11:

The table shows the values of a function obtained from an experiment. Estimate 𝑓 ( 𝑥 ) 𝑥 d using three equal subintervals with midpoints.

𝑥 1 5 9 13 17 21 25
𝑓 ( 𝑥 ) 4 2 . 7 2 0 . 8 0.2 0.9 1.5

Q12:

The table shows the values of a function obtained from an experiment. Estimate 𝑓 ( 𝑥 ) 𝑥 d using three equal subintervals with midpoints.

𝑥 4 5 6 7 8 9 10
𝑓 ( 𝑥 ) 2 . 9 2 . 4 1 . 9 1 . 1 0.1 1.1 1.7

Q13:

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

Q14:

Given 𝑓 ( 𝑥 ) = 2 𝑥 4 2 and 1 𝑥 5 , evaluate the Riemann sum for 𝑓 with six subintervals, taking sample points to be midpoints.

  • A38
  • B 5 9
  • C86
  • D59
  • E 8 6

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