### Create a StudySoup account

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

Already have a StudySoup account? Login here

# Class Note for MATH 2433 at UH

### View Full Document

## 16

## 0

## Popular in Course

## Popular in Department

This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

## Popular in Subject

## Reviews for Class Note for MATH 2433 at UH

### 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: 02/06/15

1 Applications 1 Volume lf fXy 20 on then V solid cylinder that has the surface Z fX y as its top sides 113 Area ldXdyarea of Q 111 Mass of a Plate If the density of a plate at a point Xy in the closed bounded region is given by a continuous function X then the mass of the plate is MA wwww iv Center of Mass of a Plate Let the continuous function X be the density function of a plate Then the coordinates XM yM of the center of mass of the plate are given by fX y dXdy is the volume of the Q as its base and vertical XM dXdy yM Where M is the mass of the plate ywwww LLTTLL ll TRIPLE INTEGRALS a De nition Let f fXy Z be continuous on the box T a1 X az b1 y bz 012 02 Let 73 be a partition of T and let 117 and lm be the minimum and maximum values of f on the jjk subebox Rigk Then n m l 1 Lower sum LAP mm Xi yj Zk i1j1k1 m l 11 Upper sum Uf73 2 Mm Xi y Zk i1j1k1 n m l 111 Riemann sum SAP fX y fk Xi y Zk Where 4922 is 111 k1 a point in Rigk The t ple integral of f over T is the unique number I that satis es LAP S S Uf73 Notation I fXyZdXdde T Let for all partitions 73 be an arbitrary closed bounded region in space Then fXyZdXdyZAFXyZdXdde Where T is a box that contains on T 7 and FXyz fXyZ on and FXy 0 b Repeated Integrals If the region is given by agxg b 1ltXgt y zoo my 2 my Type I region then b 152w 2y fXy Z dXdde fX y Z dz 1de Q a 1w INN Note There are ve more types of special regions c Cylindrical Coordinates QfXyZdXddeAfTCOS rsin Zrdrd dz CHAPTER 17 LINE INTEGRALS Given a vector eld FXy PXyi QXyj and a smooth or piecewise smooth curve C C X Xu y yu a g u g b parametric form C ru Xu i yuj a g u g b vector form Or in three dimensions a vector eld FX y Z PXy Z i QX y Zj RXy Z bfk and a smooth or piecewise smooth curve C C X Xu y yu Z 2a a g u g b parametric form C ru Xu i yuj Zu k a g u g b vector form DEFINITION The line integral of F over C is the number given by b Fra r Frur udu C a Alternative notations Fr a r PXy dX QXy dy FT d5 2 dimensions 0 C C 01quot Fr dr PXy Z dX QX y Z dy RXy Z dz FT d5 3 dimensions 0 C C Where FT is the component of F on the unit tangent vector T THEOREM Line integrals are invariant under orientationepreserving changes of parameter THEOREM Reversing the orientation of C changes the sign of the integral LCFra r7CFrdr cf fX 1X 7 fX dX FUNDAMENTAL THEOREM OF LINE INTEGRALS Given a curve C ru a g u g b and a vector eld F If F Vf for some function fX then C Fr a r fB 7 fA Where A ra and B rb DEFINITION The curve C is closed if ra rb COROLLARY 1 If F is the gradient of some function f and the curve C is closed then CFrdrO COROLLARY 2 Independence of Path If F is the gradient of some function f and if C1 and C2 are any two curves Which begin at A and end at B then Frdr Frdr 01 02 DEFINITION The curve C is a simple closed curve if it is closed and rt1 7 rt2 for all a lt Q lt Q lt b The positive direction on C is counterclockwise A simple closed curve is also called a Jordan curve The region enclosed by a simple closed curve is called a Jordan region GREEN S THEOREM Given a simple closed curve C oriented in the counterclockwise direction and a vector eld FXy PXy i QXyj pxydxoxydyQjof p 7 dXd yl y Where is the Jordan region enclosed by C COROLLARY TO GREEN S THEOREM If C is a simple closed curve and if j g that is if F PXy i QXyj is a gradient then y PXydXQXdy0 AREA OF USING GREEN S THEOREM C a simple closed curve enclosing the region Area of ldXdyf idef Xdy ideXdy o c c 2 c SUMMARY CHAPTERS 16 and 17 CHAPTER 16 MULTIPLE INTEGRALS I DOUBLE INTEGRALS a De nition Let f fXy be continuous on the rectangle R a g X g b c g y g 1 Let 73 be a partition of R and let m and IVA7 be the minimum and maximum values of f on the Lj suberectangle R Then M3 M3 1 Lower sum Lf73 ml Xi M 1 1739 M3 M3 113 Upper sum Uf73 Mj Xi M 1 W Z fXy X1 yj Where ngg is a point in R j1 13911 Riemann sum Sf73 l H The double integral of f over R is the unique number I that satis es Lf73 g I g Uf73 for all partitions 73 Notation I fXydXdy R Let be an arbitrary closed bounded region in the plane Then 9 fXydXdyRFXy dXdy Where R is a rectangle that contains and F X fX y on and F X y 0 on R7 b Repeated Integrals If the region is given by a g X b 1X g yg 2X Type I region then b 152m fXy dXdy fXy 1de 9 a 151m If the region is given by c g yg d 10 g X 20 Type II region then 11 2y fXy dXdy fXy WOW 9 0 19 c Polar Coordinates fXydXdy fr cos rsin rdrd r2 r2

### 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

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

#### "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."

#### "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."

### 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

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.