Back{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
∫₁⁴ 2√𝓍 d𝓍
{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.
∫₁⁴ 2√𝓍 d𝓍
{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
∫₀¹ cos ⁻¹ 𝓍 d𝓍
{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(a) Write the left and right Riemann sums in sigma notation for an arbitrary value of n.
∫₀¹ cos ⁻¹ 𝓍 d𝓍
{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
∫₀¹ (𝓍² + 1) d𝓍
The following functions are positive and negative on the given interval.
ƒ(𝓍) = xe⁻ˣ on [-1,1]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.
{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.
f(x) = sin 2x on [0,3π/4]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.
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 = ________ .
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} 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].