Q.62 Prove that 鈭�13(3x+4)聽dx=20聽i... [FREE SOLUTION]

Chapter 4: Q.62 (page 354)

Prove that ${鈭�}_{1}^{3} (3 x + 4) d x = 20$ in three different ways:
(a) algebraically, by calculating a limit of Riemann sums;
(b) geometrically, by recognizing the region in question as a trapezoid and calculating its area;
(c) with formulas, by using properties and formulas of definite integrals.

Short Answer

Expert verified

$(a) {鈭�}_{1}^{3} (3 x + 4) d x = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} [3 (1 + \frac{2 k}{n}) + 4] (\frac{2}{n}) = \lim_{n 鈫� 鈭�} (\frac{14}{n}) {鈭�}_{k = 1}^{n} 1 + \lim_{n 鈫� 鈭�} (\frac{12}{n^{2}}) {鈭�}_{k = 1}^{n} k = \lim_{n 鈫� 鈭�} (\frac{14}{n}) n + \lim_{n 鈫� 鈭�} (\frac{12}{n^{2}}) [\frac{n (n + 1)}{2}] = 14 + \frac{12}{2} = 20$

$(b) {鈭�}_{1}^{3} (3 x + 4) d x = 3 \frac{1}{2} (3^{2} - 1^{2}) + 4 (3 - 1) = \frac{3}{2} (8) + 4 (2) = 20$

$(c) {鈭�}_{1}^{3} (3 x + 4) d x = {鈭�}_{1}^{3} 3 x d x + {鈭�}_{1}^{3} 4 d x = 3 (\frac{1}{2}) (3^{2} - 1^{2}) + 4 (3 - 1) = 12 + 8 = 20$

Step by step solution

Step 1. Given information.

The given integral is ${鈭�}_{1}^{3} (3 x + 4) d x = 20 .$

Step 2. Part (a).

Take the interval $[1, 3] .$

$鈭� x = \frac{3 - 1}{n} = \frac{2}{n} x_{k} = a + k 鈭� x x_{k} = 1 + k (\frac{2}{n})$

Use Riemann's sum.

localid="1648582598852" ${鈭�}_{1}^{3} (3 x + 4) d x = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} f (\overset{*}{x_{k}}) 鈭� x = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} [3 (1 + \frac{2 k}{n}) + 4] (\frac{2}{n}) = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} (7 + \frac{6 k}{n}) (\frac{2}{n}) = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} (\frac{14}{n}) + \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} (\frac{12 k}{n^{2}}) = \lim_{n 鈫� 鈭�} (\frac{14}{n}) {鈭�}_{k = 1}^{n} 1 + \lim_{n 鈫� 鈭�} (\frac{12}{n^{2}}) {鈭�}_{k = 1}^{n} k = \lim_{n 鈫� 鈭�} (\frac{14}{n}) n + \lim_{n 鈫� 鈭�} (\frac{12}{n^{2}}) [\frac{n (n + 1)}{2}] = \lim_{n 鈫� 鈭�} 14 + \lim_{n 鈫� 鈭�} [\frac{12 n^{2} + 12}{n^{2}}] = 14 + \frac{12}{2} = 20$

Step 2. Part (b).

The limit is the area covered by the trapezoid of widths a and b and height $b - a .$

$\frac{a + b}{2} (b - a) = \frac{1}{2} (b^{2} - a^{2})$

SO the area in the interval $[1, 3]$ is

localid="1648583331166" ${鈭�}_{1}^{3} (3 x + 4) d x = 3 \frac{1}{2} (3^{2} - 1^{2}) + 4 (3 - 1) = \frac{3}{2} (8) + 4 (2) = 12 + 8 = 20$

Step 4. Part (c)

use properties and formulas of definite integrals.

${鈭�}_{1}^{3} (3 x + 4) d x = {鈭�}_{1}^{3} 3 x d x + {鈭�}_{1}^{3} 4 d x = 3 (\frac{1}{2}) (3^{2} - 1^{2}) + 4 (3 - 1) = 3 (4) + 4 (2) = 12 + 8 = 20$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Part (a).

Step 2. Part (b).

Step 4. Part (c)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Pure Maths

Statistics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Prove that 鈭�13(3x+4)dx=20in three different ways:(a) algebraically, by calculating a limit of Riemann sums;(b) geometrically, by recognizing the region in question as a trapezoid and calculating its area;(c) with formulas, by using properties and formulas of definite integrals.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Part (a).

Step 2. Part (b).

Step 4. Part (c)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Pure Maths

Statistics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.