Worksheet: Numerical Integration: Riemann Sums

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 𝑓(π‘₯)π‘₯d using three equal subintervals with left endpoints.

π‘₯57911131517
𝑓(π‘₯)βˆ’3βˆ’1.7βˆ’0.60.41.82.53.1

Q2:

Using the midpoint rule with 𝑛=5, round ο„Έ2π‘₯3π‘₯+2π‘₯d to four decimal places.

Q3:

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

π‘₯1234567
𝑓(π‘₯)βˆ’3.3βˆ’2.1βˆ’1.3βˆ’0.10.92.13.1

Q4:

Estimateο„Έ5ο€»2√3π‘₯π‘₯sind using the midpoint rule with 𝑛=4, giving your answer to four decimal places.

Q5:

Estimate ο„Έβˆ’4π‘₯(5π‘₯)π‘₯οŠ¦οŠ±οŽ„οŠ¨sind 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„Έ5√2π‘₯+1π‘₯d. Give your answer to four decimal places.

Q7:

Approximate the integral ο„Έο€Ή3π‘₯βˆ’5π‘₯π‘₯d using a Riemann sum with right endpoints. Take the number of rectangles 𝑛 to be 8.

  • A634
  • Bβˆ’634
  • C232
  • D432
  • Eβˆ’232

Q8:

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

π‘₯371115192327
𝑓(π‘₯)βˆ’2.6βˆ’1.4βˆ’0.70.82.33.44.8

Q9:

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

  • A16, an overestimate
  • B16, an underestimate
  • C48, 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 𝑓(π‘₯)π‘₯d.

π‘₯101316192225
𝑓(π‘₯)βˆ’11βˆ’314810
  • Alower bound: βˆ’20, upper bound: 1
  • Blower bound: βˆ’1, upper bound: 20
  • Clower bound: βˆ’60, upper bound: 3
  • Dlower bound: βˆ’3, upper bound: 60
  • Elower bound: βˆ’12, upper bound: 66

Q11:

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

  • A48, an underestimate
  • B18, an underestimate
  • C16, an underestimate
  • D16, an overestimate
  • E48, an overestimate

Q12:

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

  • A16, an overestimate
  • B13, an underestimate
  • C48, an underestimate
  • D16, an underestimate
  • E48, an overestimate

Q13:

Suppose that 𝐹′(π‘₯)=3οŠ±ο—οŽ‘ 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
  • D7.769, 7.686
  • E7.686, 7.769

Q14:

Calculate the right endpoint estimate of 𝑒π‘₯οŽ οŽ‘οŽ’οŽ‘οŠ¦οŠ¨ο—οŠ°οŠ«ο—οŠ°οŠͺd with 8 subintervals of equal width. Give your answer to 2 decimal places. State whether your estimate is an overestimate or underestimate of the actual value of the integral.

  • A2.77, an underestimate
  • B2.77, an overestimate
  • C4.78, an overestimate
  • D3.61, an underestimate
  • E4.78, an underestimate

Q15:

Calculate the left endpoint estimate of ο„Έπ‘₯+3π‘₯+5π‘₯d with 8 subintervals of equal width.

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