Learning starts with a question
First time here? Checkout the FAQs!
x

*Math Image Search only works best with zoomed in and well cropped math screenshots. Check DEMO

0 like 0 dislike
17 views
Approximate \(\int_0^1 e^{-x^2} d x\) to two decimal places.
in Mathematics by Platinum (141,884 points) | 17 views

1 Answer

0 like 0 dislike
Best answer
The second derivative of \(f=e^{-x^2}\) is \(\left(4 x^2-2\right) e^{-x^2}\), and it is not hard to see that on \([0,1],\left|\left(4 x^2-2\right) e^{-x^2}\right| \leq 2\). We begin by estimating the number of subintervals we are likely to need. To get two decimal places of accuracy, we will certainly need \(E(\Delta x)<0.005\) or
\[
\begin{aligned}
\frac{1}{12}(2) \frac{1}{n^2} &<0.005 \\
\frac{1}{6}(200) &<n^2 \\
5.77 \approx \sqrt{\frac{100}{3}} &<n
\end{aligned}
\]
With \(n=6\), the error estimate is thus \(1 / 6^3<0.0047\). We compute the trapezoid approximation for six intervals:
\[
\left(\frac{f(0)}{2}+f(1 / 6)+f(2 / 6)+\cdots+f(5 / 6)+\frac{f(1)}{2}\right) \frac{1}{6} \approx 0.74512 .
\]
So the true value of the integral is between \(0.74512-0.0047=0.74042\) and \(0.74512+\) \(0.0047=0.74982\). Unfortunately, the first rounds to \(0.74\) and the second rounds to \(0.75\), so we can't be sure of the correct value in the second decimal place; we need to pick a larger \(n\). As it turns out, we need to go to \(n=12\) to get two bounds that both round to the same value, which turns out to be \(0.75\). For comparison, using 12 rectangles to approximate the area gives \(0.7727\), which is considerably less accurate than the approximation using six trapezoids.

In practice it generally pays to start by requiring better than the maximum possible error; for example, we might have initially required \(E(\Delta x)<0.001\), or
\[
\begin{aligned}
\frac{1}{12}(2) \frac{1}{n^2} &<0.001 \\
\frac{1}{6}(1000) &<n^2 \\
12.91 \approx \sqrt{\frac{500}{3}} &<n
\end{aligned}
\]
Had we immediately tried \(n=13\) this would have given us the desired answer.
by Platinum (141,884 points)

Related questions

0 like 0 dislike
0 answers
asked Jan 13 in Mathematics by MathsGee Platinum (141,884 points) | 85 views
0 like 0 dislike
1 answer
1 like 0 dislike
0 answers
1 like 0 dislike
1 answer
asked Apr 26, 2020 in Mathematics by MathsGee Platinum (141,884 points) | 145 views
0 like 0 dislike
1 answer
asked Feb 4 in Mathematics by Gauss Diamond (66,747 points) | 116 views

Join the MathsGee Q&A forum where you get STEM education support to succeed from our community. Connect and Learn.


On the MathsGee Q&A Forum, you can:


1. Ask questions


2. Answer questions


3. Vote on questions and answers


4. Start a fundraiser


5. Tip your favorite community members


6. Create Live Video Tutorials (Paid/Free)


7. Join Live Video Tutorials (Paid/Free)


8. Earn points by participating



MathsGee Q&A forum post


1. Remember the human


2. Act like you would in real life


3. Find original source of content


4. Check for duplicates before publishing


5. Read the community guidelines




FORUM RULES


1. Answers to questions will be posted immediately after moderation


2. Questions will be queued for posting immediately after moderation


3. Depending on the number of messages we receive, you could wait up to 24 hours for your message to appear. But be patient as posts will appear after passing our moderation.




USEFUL LINKS


Acalytica | Web Analytics | SEO Reports | Social Proof Tool | Email Marketing


MathsGee Android Q&A