Identifying Riemann sums Fill in the blanks with an interval and a value of n.


4
∑ ƒ (1 + k) • 1 is a right Riemann sum for f on the interval [ ___ , ___ ] with
k = 1
n = ________ .

{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.

(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.

∫₀⁴ (4𝓍― 𝓍²) d𝓍

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function ƒ on [a,b]. Identify ƒ and express the limit as a definite integral.                                

          n                                                                                                                                                                              

    lim   ∑ (𝓍ₖ*² + 1) ∆𝓍ₖ on [0,2]                                                                                                                                                                            

  ∆ → 0   k=1                                                                                                                                                                                                                      

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function ƒ on [a,b]. Identify ƒ and express the limit as a definite integral.                                

          n                                                                                                                                                                              

    lim   ∑   𝓍*ₖ (ln 𝓍*ₖ) ∆𝓍ₖ on [1,2]                                                                                                                                                                            

  ∆ → 0   k=1                                                                                                                                                                                                                      

Identifying Riemann sums Fill in the blanks with an interval and a value of n.

4

∑ ƒ (1.5 + k) • 1 is a midpoint Riemann sum for f on the interval [ ___ , ___ ]

k = 1

with n = ________ .

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The midpoint Riemann sum for f(x) = x³ on [3,11] with n = 32.

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The right Riemann sum for ƒ(𝓍)) = x + 1 on [0, 4] with n = 50.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(c) For an increasing or decreasing nonconstant function on an interval [a,b] and a given value of n, the value of the midpoint Riemann sum always lies between the values of the left and right Riemann sums.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(b) A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a,b].