×

### Let's log you in.

or

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

×

or

## Real Analysis

by: Otilia Murray I

111

0

11

# Real Analysis MAT 125A

Otilia Murray I
UCD
GPA 3.88

Qinglan Xia

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

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

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

×
Unlock Preview

COURSE
PROF.
Qinglan Xia
TYPE
Class Notes
PAGES
11
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 11 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 125A at University of California - Davis taught by Qinglan Xia in Fall. Since its upload, it has received 111 views. For similar materials see /class/187370/mat-125a-university-of-california-davis in Mathematics (M) at University of California - Davis.

×

## Reviews for Real Analysis

×

×

### What is Karma?

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

Date Created: 09/08/15
56 Substition and area between curves Last time we knew that for inde nite integral we have the substitution rule fltgltxgtgtg ltxgtdx fudu ifwe let u How about de nite integral b 91 fgxgxdx fudU a g a Why Let F be any antiderivative of f Then F is an antiderivative of fg So by the Fundamental theorem of Calculus b fgmgxdm Fltgltxgtgt 91 Fltgltbgtgt Fltgltagtgt FltugtIEZ fudU Remark There are two methods to evaluate fg xdx 0 apply the substitution rule for de nite integral 5 95 fltgltxgtgtg mac fltugtdu a 9a 0 or apply the substitution rule for inde nite integral fgxgxdx fudu some function ofx C Then substitute 05 a x b in the end Which way is better Depends on the particular problem 0 Advantage for the rst one don t need to replace u by o Advantage for the second one don t need to calculate endpoints for each substitution step Example Evaluate fo 13 dx 21 12l W 205 du 3 3 9 d 8 23u8221 0 052 1W3 m 1 1113 2 I 1 2 gt 2 Method 2We apply the substitution rule for inde nite integral by letting u x2 1 2f i du 73 23 73 2 23 x2 1 W i 2 Therefore 295 d 3 2 Jr 123z 3823 123 3 3 9 a a 0 x2 1W3 2 10 2 2 2 64 Example fag 31 Let u ln x then 4 6 dx i 4 du 62 xlnxln nx i 2 ulnu i ln4 ln2 t 111211333 1 1 4 1110112 1 m4 n n 11 ln2 ln21n2ln2 ln2 Example What is 2 sinyc5 303cm 2 Seems not that easy to evaluate But indeed the value is simply 0 Why Here the integrand 990 sinyc5 353 is an odd function on a symmetric interval 2 2 Let f be a function on a symmetric interval a a o fyc is an odd function if f 96 f96 o fyc is an even function if f ac Check g yc sin yc5 yc3 sin 305 303 g3c so 9 is odd De nite integrals of symmetric functions Theorem Let f be continuous on a symmetric interval a a foam 20afdm o If f is even then o ff is odd then fxdx 0 Example f32sinx5 mgldx 0 because sin 955 x3 is an odd function on a symmetric interval 2 2 Proof of the theorem Assume f is odd then mm fxdx0afxdx But 0 7a fltxgtdx fltxgtdx 7a 0 f udu by using substitution u x 0 fudu since f is an odd function 0 fudu 0 Therefore a a a fxdac fudu fxdx 0 7a 0 0 Example 5 3 1001 955 39 2 t 1 d st1n x7T4 an4 x lac 5 3 1001 955 5 sin 205 tan xldx0 ldx10 Ll 74 lt4 gt l 75 1001 This is because sin3 2x I tan x is an odd function on a symmetric interval 5 5 V7r4 Example qu hdx Note that f 4 is an even function on a symmetric interval So M M2Am m 2A3x 4dx 2 2i 495l3 9 2E 129 24 15 Note ffa xdx 0 as x is an odd function on a symmetric interval Area between curves 0 If f 2 0 is integrable on a b then the area between the graph of f and the XaXis is ab Suppose M 2 gm 2 o then the area between f and 990 is fab fltxgtdw bgltwgtdw o How about f 2 935 without the condition that 990 2 0 9 39 2quot 3 I 3 3 E H Note that we may always pick a large constant M so that fM29MZO Thus the area is b b me M gm Mldw W gwldw De nition If f and g are continuous function with f 2 930 on a b then the area of the region between the curves y f3c andy g30from a to b is b A W gltwgtdw What happens if we do not require f 2 930 The area between curves is b A inn gem Example Find the area of the region enclosed by the parabola y 2 902 and the line y yc Step 1 Find the intersection points of these two curves That is we solve y w y 2 302 simultaneously for ac yc2 302 302 30 20 30130 20 1 122 So the intersection points are P1 1 1 and 1322 2 On 1 2 the curve y 2 902 is above the line y 30 Therefore Step 2 Find the area Area 12 302 ycdyc 2 x2wd 1 303 302 L i2 8 1 1 4 22 3 321 6 32 1 41 2 239 Example Find the area of the region in the rst quadrant that is bounded above by y and below by the acaXis and y 902 14 Intersection points are 0 014 0 and 4 2 So the area is Method 1 fo dw 4 302 14dyc Method 2 4 4 dw 302 l4dyc 0 2 y14 y2dy 0 or method 3 integration with respect to y In general for regions like we may nd the area between two curves by integrating with respect to y instead of 90 d A my gltygtgtdy Example Find the area of the region in the rst quadrant that is bounded above by the XaXis and below by y 2352 and 30 y 3 Intersection points are 0 0 3 0 and 1 2 The area is Method 1 Integration with respect to 35 1 3 8 22d 3 acdyc 0 1 3 Method 2 Integration with respect to y

×

×

### BOOM! Enjoy Your Free Notes!

×

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

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."

Kyle Maynard Purdue

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made \$280 on my first study guide!"

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."

Parker Thompson 500 Startups

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION 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 support@studysoup.com

#### STUDYSOUP REFUND POLICY

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: support@studysoup.com

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 support@studysoup.com