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 III

by: Alvena McDermott

Calculus III MATH 2433

Alvena McDermott
GPA 3.69


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

Class Notes
25 ?




Popular in Course

Popular in Mathmatics

This 5 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 2433 at University of Houston taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/208411/math-2433-university-of-houston in Mathmatics at University of Houston.


Reviews for Calculus III


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
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 1 Lower sum Lf73 szi39 Xi M 113 Upper sum Uf73 ZZMj Xi M M 111 Riemann sum Sf73 fXy X1 yj Where ngg isa point in R i1 391 u 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 fX y dXdy fX y dXdy 9 0 1PM c Polar Coordinates fXydXdy fr cos rsin rdrd r2 r2 1 Applications 13 11 111 IV Volume lf fXy Z 0 on then V fXy dXdy is the volume of the 9 solid cylinder that has the surface Z fX y as its top sides as its base and vertical Area 1 dXdy area of 9 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 M X y dXdy 9 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 QX XydXdy Qy XydXdy T yMT Where M is the mass of the plate XM 11 TRIPLE INTEGRALS a De nition Let f fX y Z be continuous on the box T a1 X az b1 y bz 012 02 Let 73 be a partition of T and let mijk 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 n m l 1139 Upper sum Uf73 2 Mjk Xi yj Zk i1g1k1 n m l 111 Riemann sum SAP fXyfk Xi y Zk Where 4922 is i1j1k1 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 fX y Z dXdde T Let be an arbitrary closed bounded region in space Then fXyZdXdyZAFXyZdXdde and FXy Z fXy Z on for all partitions 73 Where T is a box that contains and F X y 0 onTi b Repeated Integrals If the region is given by altxlt 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 PXy 391 QXyJ39 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 lt 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 Fr a r o C 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 ClFra rC2Frdr 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 PXydXQXJydyQ7Qiil dXdy Where is the Jordan region enclosed by C COROLLARY TO GREEN S THEOREM If C is a simple closed curve and if 7 70 that is if F PXy i QXyj is a gradient then 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


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."

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"


"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.