Worksheet: Riemann Sum

In this worksheet, we will practice finding and comparing the areas under the curves of functions using the Riemann sum.

Q1:

Calculate the midpoint rule estimate of 𝑥 + 2 𝑥 4 0 2 d with 𝑛 = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A16, an underestimate
  • B48, an overestimate
  • C28, an overestimate
  • D28, an underestimate
  • E16, an overestimate

Q2:

The table gives sampled values of an increasing function 𝑓 . Use the data to give a lower and upper bound for 𝑓 ( 𝑥 ) 𝑥 d .

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

Q3:

Calculate the left endpoint estimate of 𝑥 + 2 𝑥 4 0 2 d with 𝑛 = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A16, an overestimate
  • B18, an underestimate
  • C48, an underestimate
  • D16, an underestimate
  • E48, an overestimate

Q4:

Calculate the right endpoint estimate of 𝑥 + 2 𝑥 4 0 2 d with 𝑛 = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A48, an underestimate
  • B16, an underestimate
  • C16, an overestimate
  • D48, an overestimate
  • E13, an underestimate

Q5:

Suppose that 𝐹 ( 𝑥 ) = 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.

  • A7.769, 7.843
  • B7.843, 7.686
  • C0.7689, 0.686
  • D7.769, 7.686
  • E7.686, 7.769

Q6:

The table gives sampled values of an increasing function 𝑓 . Use the data to give a lower and upper bound for 𝑓 ( 𝑥 ) 𝑥 d .

𝑥 12 14 16 18 20 22
𝑓 ( 𝑥 ) 1 4 1 1 5 1 6 15
  • Alower bound: 5 2 , upper bound: 20
  • Blower bound: 1 2 , upper bound: 46
  • Clower bound: 2 3 , upper bound: 6
  • Dlower bound: 4 6 , upper bound: 12
  • Elower bound: 6 , upper bound: 23

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.