New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

Be part of our community, it's free to join!

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

Calculus II

by: Alvena McDermott

Calculus II MATH 1432

Alvena McDermott
GPA 3.69

Jiwen He

Almost Ready


These notes were just uploaded, and will be ready to view shortly.

Purchase these notes here, or revisit this page.

Either way, we'll remind you when they're ready :)

Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

Unlock Preview
Unlock Preview

Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

View Preview

About this Document

Jiwen He
Class Notes
25 ?




Popular in Course

Popular in Mathmatics

This 9 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1432 at University of Houston taught by Jiwen He in Fall. Since its upload, it has received 30 views. For similar materials see /class/208374/math-1432-university-of-houston in Mathmatics at University of Houston.


Reviews for Calculus II


Report this Material


What is Karma?


Karma is the currency of StudySoup.

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/19/15
LGCture 1 lsection 87 Numerical Integration J iwen He 1 Riemann Sums 11 Area Problem Area Problem 1 V I 9192 93 9n a b x a x0 x1 x2 x3 x b x Partition of a 9 Take a partition P 2 130561 xn of ab Then P splits up the interval a 9 into a nite number of subintervals 5130 131 58n17513n with a 80 lt 5131 lt lt Slim 2 b We have ab 51305131 U U cat451 U U CEn1CEn Remark This breaks up the region 9 into n subregions 21 Qn We can estimate the total area Off by estimating the area of each sabregion 21 and adding up the results 12 Lower and Upper Sums Lower and Upper Sums m I 1 I I I II l I l I I I l I l l I Let mfg 13141 mi 2 Qz luUQiUH UQn raftAsa 2 area of rz 3 area of 21 g div area of shaded region is a area of shaded region is an lower sum forf upper sum forf n b n i1 a i1 1 3 Riemann Sum Riemann Sum 12 y y v 1 X I 1 X2 X3 X4 X5 A X6 X7 8 l area of shaded region Is a 0 1 2 3 4 5 6 quot7 8 area of shaded region is an lower sum forf upper sum forf Riemann Sum mm fiA1 fv Av2 room where is any point picked in 1371i for z39 l n We have L141 Emmi g SP Z mam g ZMiAxi 2 Ufa i1 i1 i1 Limit of Riemann Sums b foe dx I 113180 room fv Av2 mama 2 Rectangle Approximations 2 1 Regular Partition Regular Partition A V Problem 2 d Find the approximate value of ln 2 f 1 It 6 7 8 9 1 th 1 f l t 1 2 uslngony evaueso fz ma 7 5 5 5 57 Numerical Integration Approximate f f d3 using only values of f at n 1 equally spaced points between a and 17 Regular Partition of a 17 Let 33 a l iALE 73 01 n Where A3 b Ta Then ab 130331 U U 3371 13 U 22 Approximations Rectangle Approximations X 1 W AL xil x X 26171 Xi X Xi1 Xi X f flta2gt da fltai1gtAa fezma f xquot 12xquotAa xi l Rectangle Approximations of fjil f d3 fezAme mam frrm leftendpoints rightendpoints midpoints Rectangle Approximations of f f d3 21 xi f d3 i 1 Efrem Zitagtva 2 Am i 1 1 7 21 left endpoints right endpoints midpoints Rectangle Rules b Rectangle Rule for Approximating fa d9 0 Left endpoint rule Ln b a f90 f91 fan1 b o nght endpomt rule Rn W1 fw2 mm o Midpoint rule Mn 7 b a 11 937171 937 n 2 quot39fTl y x Problem Find the approximate value of 1112 12 d 118ng Only the Values Of f l at l g g g 2 Approximating 1112 169314718 51s07456 RBf ltgg g gt 06456 M5 l g 02 gtm06919 23 Error Estimates Error Estimates 7 Nb 7777 gt Ti l I gt I I I W J I I I I I I I I Nb arm I I I i ii gt I I I I I I I Ha gt Pg u 7 r each width is Iquot here 2 6 b b a IImIdm Ln 3 n 1 fa Error Estimates b 1 b Left endpoint rule E5 f d3 Ln 6 fc 0Am a n b 1 b Right endpoint rule E5 f d3 Rn 6 fc 0Am a n M 7 b 7 i b 03 7 2 o Mldpomt rule En 7 d3 Mn 7 24 2 f c i 0Am a n Error Estimates Example Error Estimates b 1 b a2 o Left endpoint rule E7 d3 Ln 5 fc 0Aac a n b 1 b a2 Right endpoint rule E5 f d3 Rn 5 fc 0Am a n M 7 b 7 i b a 7 2 o Mldpomt rule En i d3 Mn 7 24 c 0Am a n A Approximating In 2 069314718 Note that 7 E127 fl 6 2 1 2 1E511E 1 lt 5 maXce12if Ci 1 10 2 1 3 S 52 maXCE12 LINCM W10 3 Trapeziodal and Parabolic Approximations 3 1 Approximations Trapeziodal and Parabolic Approximations V 7 f f f 9 I Xi1 quotIi 1 X x x1 x x xi l Xi x 2 f5quot 1 f d3 Trapeziodal Parabolic L Approximations of fig fda fi 1 fi Am fz 1 4f xi 12xi fag A51 trapezioa al Parabolic Approximations of f fa d3 2121 3 fz d3 2 1fltxi1gt 11 7 2121 Mail 4f 5 125 quot ma T rapezioa al Parabolic Trapeziodal and Simpson s Rules b Trapeziodal and Simpson s Rules for Approximating fa d9 0 Trapeziodal Rule Tn b2 na f930 2f931 2f93n71 b o Simpson7s Rule Parabolic Sn 6 na f9o 2 2fan1 wwwwwn y x Problem Find the approximate value of ln2 12 If using only the values of f i at 1 E I 2 2 7 57 57 57 57 39 Approximating ln2 T 069371816 10 REUF7 Jr Sse12lt2 10 10 10 10 10 MHETTH 32 Error Estimates Error Estimates 1717 fa P bia fb ifu biu each width is here 2 l1 f mw n Error Estimates b o Trapeziodal Rule E7 lt 2 1 b a n 6 elm Tn b o Simpsorfs Rule ES clan Sn Error Estimates Example Error Estimates b o Trapeziodal Rule E7 elm Tn b o Simpsorfs Rule ES clan Sn 12 1 b a5 2880 12 1 b a5 2880 gtU f 19 ommm 19 ommm f4c 0A4 f4c 0A4 x Approximating ln2 069314718 Note that faj m 123 f 9 7 f3L39 m 47 f493 271 IEEI s 1 1 52gt maXce12lfCl 271 5 IE S FEM 54 maXcem f40 Outline Contents 1 Riemann Sums 1 11 Area Problem 1 12 Lower and Upper Sums 1 13 Riemann Sum 2 2 Rectangle Approximations 2 21 Regular Partition 2 22 Approximations 3 23 Error Estimates 4 3 Trapeziodal and Parabolic Approximations 6 31 Approximations 6 32 Error Estimates 7


Buy Material

Are you sure you want to buy this material for

25 Karma

Buy Material

BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.


You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Jim McGreen Ohio University

"Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Jennifer McGill UCSF Med School

"Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over $500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

Steve Martinelli UC Los Angeles

"There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."


"Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!

Refund Policy


All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email


StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here:

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.