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Math
Calculus
Chapter 5
Fill in the blanks to complete each of the following theorem statements:

Chapter 5: Fill in the blanks to complete each of the following theorem statements: (page 495)

Q. 17
If $f$ is integrable and monotonic on $[a, b]$ and $n$ is a positive integer, then
role="math" localid="1663147207302" $| E_{L E F T} (n) | 鈮� | 鈥� - 鈥� | = |鈥�| 鈭� x$

Short Answer

Expert verified

$| E_{L E F T} (n) | 鈮� | L E F T (n) - R I G H T (n) | = |f (b) 鈥� f (a)| 鈭� x$

Step by step solution

01

Given

$f$ is integrable and monotonic on $[a, b]$ and $n$ is a positive integer,

02

Error Bounds for Left and Right Sums

Let $f$ is integrable and monotonic on $[a, b]$ and $n$ is a positive integer. Then we can bound the error $E_{L E F T} (n)$ of the left sum as follows:

$| E_{L E F T} (n) | 鈮� | L E F T (n) - R I G H T (n) | = |f (b) 鈥� f (a)| 鈭� x$

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Most popular questions from this chapter

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$鈭� \frac{x}{\sqrt{x^{2} - 1}} d x$

Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution with $x = a \sin u$ or $x = a \tan u$ ? Why do we need to think about the unit circle after trigonometric substitution with $x = a s e c u$ ?

Consider the integral $鈭� x {(x^{2} 鈭� 1)}^{2} d x$ .

(a) Solve this integral by using u-substitution.

(b) Solve the integral another way, using algebra to multiply out the integrand first.

(c) How must your two answers be related? Use algebra to prove this relationship.

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$x^{2} - 5 x + 1$

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) An integral with which we could reasonably apply trigonometric substitution with $x = \tan 鈦� u$ .

(b) An integral with which we could reasonably apply trigonometric substitution with $x = 4 \sec 鈦� u$ .

(c) An integral with which we could reasonably apply trigonometric substitution with $x 鈭� 2 = 3 \sin 鈦� u$ .

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