91Ó°ÊÓ

1. True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The left-sum and right-sum approximations are the same if the number$n$ of rectangles is very large.

(b) True or False:${∫}_{-2}^5(x+2{)}^{3}dx$ is a real number.

(c) True or False:$∫\left(5x^2-3x+2\right)dx$ is exactly 26.167.

(d) True or False: ${∫}_{-3}^{-2}f\left(x\right)dx=-{∫}_2^{3}f\left(x\right)dx$.

(e) True or False: If ${∫}_0^{2}f\left(x\right)dx=3$and ${∫}_0^{2}g\left(x\right)dx=2$, then ${∫}_0^{2}f\left(g\right(x\left)\right)=6$

(f) True or False: If ${∫}_0^{2}f\left(x\right)dx=3$and${∫}_0^{2}g\left(x\right)dx=2$, then ${∫}_0^{2}f\left(x\right)g\left(x\right)dx=6$

(g) True or False: If ${∫}_0^{1}f\left(x\right)dx=3$and ${∫}_{-1}^{0}f\left(x\right)dx=4$, then ${∫}_{-1}^{1}f\left(x\right)dx=7.$

(h) True or False: If ${∫}_0^{2}f\left(x\right)dx=3$and ${∫}_2^{4}g\left(x\right)dx=4$, then ${∫}_0^4\left(f\right(x)+g(x\left)\right)dx=7.$

Step by step solution

01

given information

$f\left(x\right)=\sqrt{x},y=0,x=0,x=3$

By using the hint we can estimate the are by calculating the limit

$A=\underset{n→∞}{\lim }\underset{i=1}{\overset{n}{∑}}f\left(c_i\right)\mathrm{Δ}{x}_i$

where $\mathrm{Δ}{x}_i$is the width of the $i$th subinterval $\left[x_{i-1},x_i\right]$, so $\mathrm{Δ}{x}_i=c_i-c_{i-1}$, hence we can calculate the sum as follows

$A=\underset{n→∞}{\lim }\underset{i=1}{\overset{n}{∑}}f\left(c_i\right)·\left[c_i-c_{i-1}\right]$

$=\underset{n→∞}{\lim }\underset{i=1}{\overset{n}{∑}}f\left(\frac{3i^2}{n^2}\right)·\left[3i^2/n^2-3(i-1{)}^2/n^2\right]$

$=\underset{n→∞}{\lim }\underset{i=1}{\overset{n}{∑}}\frac{\sqrt{3}i}{n}·\left[\frac{3i^2}{n^2}-\frac{3(i-1{)}^2}{n^2}\right]$

$=\underset{n→∞}{\lim }\underset{i=1}{\overset{n}{∑}}\frac{3\sqrt{3}i}{n^3}·\left[i^2-(i-1{)}^2\right]$

$=\underset{n→∞}{\lim }\underset{i=1}{\overset{n}{∑}}\frac{3\sqrt{3}i}{n^3}·[2i+1]$

$=3\sqrt{3}\underset{n→∞}{\lim }\frac{1}{n^3}\underset{i=1}{\overset{n}{∑}}·\left[2i^2+i\right]$

$=3\sqrt{3}\underset{n→∞}{\lim }\frac{1}{n^3}\left[2\underset{i=1}{\overset{n}{∑}}{i}^2+\underset{i=1}{\overset{n}{∑}}i\right]$

$=3\sqrt{3}\underset{n→∞}{\lim }\frac{1}{n^3}\left[2·\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\right]$

$=\sqrt{3}\underset{n→∞}{\lim }\left[\frac{n(n+1)(2n+1)}{n^3}+\frac{3n(n+1)}{2n^3}\right]$

$=\sqrt{3}[2+0]$

$=2\sqrt{3}$

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 91Ó°ÊÓ!