Chapter 4: Q. 42 (page 353)

For each definite integral in Exercises 41–46, (a) find the general n-rectangle right sum and simplify your answer with sum formulas. Then (b) use your answer to approximate the definite integral with $n = 100$ and $n = 1000$ . Finally, (c) take the limit as $n \to \infty$ to find the exact value.
$\int_{- 3}^{3} (2 x + 1) d x$

Short Answer

Expert verified

Part(a) The right sum is $\frac{36}{n^{2}} (\frac{n (n + 1)}{2}) - 30$ .

Part(b) The approximation for $n = 100$ is $6.435$ and for $n = 1000$ is $6.135$ .

Part(c) The exact value is $6$ .

Step by step solution

Part(a) Step 1. Given Information.

We are given,

$\int_{- 3}^{3} (2 x + 1) d x$

Part(a) Step 2. Finding the right sum.

The right sum defined for n rectangles on $[a, b]$ is $\sum_{k = 1}^{n} f (x_{k}) Δ x$ .

Where $Δ x = \frac{b - a}{n},$

and $x_{k} = a + k Δ x$

role="math" localid="1648740724350" $Δ x = \frac{3 + 3}{n} = \frac{6}{n}$

And,

role="math" localid="1648740860616" $x_{k} = - 3 + k (\frac{6}{n}) = \frac{6 k}{n} - 3$

Part(a) Step 3. Finding the right sum.

The right sum is given by,

$\sum_{k = 1}^{n} (2 (\frac{6 k}{n} - 3) + 1) (\frac{6}{n}) = (\frac{6}{n}) \sum_{k = 1}^{n} (\frac{12 k}{n} - 5) = (\frac{6}{n}) \sum_{k = 1}^{n} (\frac{12 k}{n} - 5) = \frac{6}{n^{2}} (12) (\frac{n (n + 1)}{2}) - \frac{6}{n} (5 n) = \frac{36}{n^{2}} (\frac{n (n + 1)}{2}) - 30$

Part(b) Step 1. Approximating the definite integral.

The right sum is,

$\frac{36}{n^{2}} (\frac{n (n + 1)}{2}) - 30$

For $n = 100$ , the approximation will be,

$= \frac{36}{100^{2}} (\frac{100 (100 + 1)}{2}) - 30 = 6.435$

For $n = 1000$ , the approximation will be,

$= \frac{36}{1000^{2}} (\frac{1000 (1000 + 1)}{2}) - 30 = 6.135$

Part(c) Step 1. Finding the exact value.

The limit is given by,

$\int_{- 3}^{3} (2 x + 1) d x = \lim_{n \to \infty} (\frac{36}{n^{2}} (\frac{n (n + 1)}{2}) - 30) = 36 - 30 = 6$

The exact value is $6$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Part(a) Step 1. Given Information.

Part(a) Step 2. Finding the right sum.

Part(a) Step 3. Finding the right sum.

Part(b) Step 1. Approximating the definite integral.

Part(c) Step 1. Finding the exact value.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Statistics

Logic and Functions

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

For each definite integral in Exercises 41–46, (a) find the general n-rectangle right sum and simplify your answer with sum formulas. Then (b) use your answer to approximate the definite integral with n=100and n=1000. Finally, (c) take the limit as n→∞to find the exact value. ∫-33(2x+1)dx

Short Answer

Step by step solution

Part(a) Step 1. Given Information.

Part(a) Step 2. Finding the right sum.

Part(a) Step 3. Finding the right sum.

Part(b) Step 1. Approximating the definite integral.

Part(c) Step 1. Finding the exact value.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Statistics

Logic and Functions

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.