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

Math 141: Exam 1 Study Guide

by: ErinNordquist

Math 141: Exam 1 Study Guide 141

Marketplace > University of Maryland > Math > 141 > Math 141 Exam 1 Study Guide

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

This study guide features: 1. Everything we have covered in class from chapter 6 separated into sections... - 6.1: Volume - 6.4: Work - 6.5: Moments and Center of Gravity - 6.8...
Calculus II
Wisely Wong
Study Guide
calculus2, calcii, calc2, Volume, Integrals, Math141, Ch.6, exam1, arclenth, Parametric Functions, work, Moments, Center of Mass, certroid
50 ?




Popular in Calculus II

Popular in Math

This 6 page Study Guide was uploaded by ErinNordquist on Monday September 12, 2016. The Study Guide belongs to 141 at University of Maryland taught by Wisely Wong in Fall 2016. Since its upload, it has received 203 views. For similar materials see Calculus II in Math at University of Maryland.


Reviews for Math 141: Exam 1 Study Guide


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: 09/12/16
Math141 Sections:6.1,6.4,6.5,6.7,6.2,6.8 ExamDate:Friday,September16,2016 Section 6.1: Volume Cross Sectional Method ● DEFINITION​: Consider solid region D for every x in an interval [a, b]. The plane perpendicular to the x-axis at x intersects D in a plane region having area A(x) with the goal of defining the volume ● ifA(x)  =  ifor alla ≤ x ≤ b meaning the area of a cross section is constant(ie. It is a shape with an area formula like a square)) then, V = A(i−a)​ , the product of constant Aiand the length of the interval. ● Riemann Sums - use when cross sections are not constant ○ P = {x 0 x1,...,nx }partitions [a, b] into n slices. Find the area of these slices to estimate the volume of D b n ○ FORMULA: ​ V = lim ∑ A(x )Δx =kA(x)kdx ∫ ||P||→k=1 a Washer Method ● DEFINITION:​ Find volume of a solid by rotating g(x) and f(x), [a, b], about x-axis for 0 ≤ g(x) ≤ f(x) ● radius = f(x) - g(x) 2 ● Area of a circle =πr 2 2 ● Area of outer function = π(f(x)) and Area of inner function = π(g(x)) ○ So because radius of washer = f(x) - g(x)... 2 2 ○ FORMULA: ​ Area of washer = π(f(x) − g(x) ) 1 b b ● FORMULA:​ V = A∫x) dx  = π(∫(x) − g(x) ) dx a a Shell Method - for rotating about y-axis ● Area of cross-section = surface area of cylinder (SA = 2πrh) ○ The x value represents the distance from the center of the cylinder (y-axis) so we use x in place of r, sA = 2πxh  ○ h is represented by the distance between f(x) and g(x), so… ○ FORMULA:​ A = 2πx(f(x) − g(x))  b b ○ FORMULA:​ V = A∫x) dx  = 2π∫(f(x) − g(x)) dx a a **Practice Problems on pg. 377 (suggested #17, 31, 33, 37 and 38) Section 6.4: Work Definition of Work ● Informally: Work = force x displacement ● DEFINITION​: consider the force of an object moving in a straight line(ie. on the x-axis) b from x = a to x = b. We define th​ ork done​ by force F(x) as W∫= F(x) dx a Hooke’s Law ● DEFINITION​: Let k be a spring constant, x is the displacement of the spring from resting position(equilibrium). The force exerted by the spring is ○ FORMULA: F(x) =− kx  Pumping a Tank ● DEFINITION​: Consider a tank filled with liquid (usually water) to height “a” where we want to pump liquid out to height “b” ○ A(x) = cross sectional area x feet from top(or pump location if not at top) 2 ○ s(x) = distance to pump out (distance water has to travel to get out of tank) ○ ω= weight per volume unit of liquid ● FORMULA​: Work ≃ (ωA(x )Δx )1(s(1 *) + 1ωA(x )Δx )2(s(2 *) + .1. + (ωA(x )Δxn) (snx*)) n ○ This is an approximation using Riemann sums to find the work b ● FORMULA: ​ W = ω ∫(x)*A(x) dx* a ○ Note: often “s(x)” is written as just “x” ○ A(x) depends on the shape of the cross section ■ Ex: Hemisphere: cross sections are circles(A(x) = πr ) 2 **Practice Problems on pg. 395 (suggested #8, 12, 15, and 24) Section 6.5: Moments and Center of Gravity Moments of Point Masses ● DEFINITION​: ​point mass​ - an object of mass “m” concentrated at the point (x, y) ● DEFINITION​: the ​moment of a point​ mass about the y-axis is defined by, “mx,” the tendency of the point mass to rotate about the y-axis ● DEFINITION​: Particles with masses M 1 M ,2.., M ​nat x1, x2,..., nrespectively are at equilibrium​ ifM x +M x +...+M x = 0​ n n  must be true for M  y M ​ x) 1 1 2 2 ● DEFINITION​: Given Particles with coordinates (x1, y1), (x2, 2 ),...,nx ,ny with masses M ,1M ,.2.,M 3 ​espectively, ​ he moment… ​ ○ About the y-axis = M y M x +M x +...+M x   n n 1 1 2 2 ○ About the x-axis = M x M y 1 1y +..2 2 y   n n ● DEFINITION:​ the ​center of mass​/gravity(centroid) of particles with masses M ,1M ,.2., M nlocated at (x1, y1), (2 , 2 ),...,nx n yis​(x, y)​so that ○ M =yM x+M1x+...2M x = Mx = n x +M x +...+M1 1  2 2 n n ○ M =xM y+M1y+...2M y = My = n y +M y +...+M1 1  2 2 n n M ○ FORMULA:​ x = y ​and​ y = M x M total M total 3 Moments of Thin Sheets(Laminas) ● DEFINITION:​ Consider a thin sheet of material such that the mass is equally distributed. We approximate a thin sheet(lamina) created by two continuous functions, f(x) and g(x), on [a, b]. ● Consider a vertical strip such that… n n ○ M y ∑(mass)(distance to y−axis) = ∑(f(x )kg(x )Δk )k k k=1 k=1 ○ Because mass is uniformly distributed, mass = M = area b ■ FORMULA:​ M = A = f(x) − g(x) dx ∫ a M y M y ■ FORMULA:​ x = M = A total Mx M x ■ FORMULA​ : y = M = A total b ○ FORMULA: ​ M =yx(∫(x) − g(x)) dx a b 1 2 2 ○ FORMULA​: M =x ∫ 2(f(x) − g(x) ) dx a Symmetry Principle ● DEFINITION:​ if a region is symmetric about a line, the centroid lies on that line ○ Note: This is often used on tests for one of the dimensions(x or y) because it cuts the amount of work in half thus enables for quicker testing. **Almost always on the ​Final​** **Practice Problems on pg. 405 (suggested #1, 6, 9, 12, 17, 24, and 32) Section 6.7: Parametrized Curves An Introduction(all we need in this course) ● DEFINITION​: Given a curve on the x-y plane, we plot points (x, y) by defining functions using the parameter “t”... 4 ○ x = x(t) = f(t) ○ y = y(t) = g(t) ● General example: x(t) = acos(t)and y(t) = asin(twhere “a” is a constant x y ○ Rewrite as cos(t) = a and sin(t) =a  2 2 ○ Remember the Trigonometric identity: cos (x)+sin (x) = 1  2 y2 ○ So a2+ a2 = 1. . . ○ FORMULA: x +y = a  2 ○ This parametrizes circle of radius “a” centered at the origin ● “Trivial Parametrization” ○ To parametrize y = f(x)..letx = t​so y = f(t) **Practice Problems on pg. 415 (suggested 1, 2, 4, 6, and 8) Section 6.8, 6.2: Arc Length Mean Value Theorem Review ● DEFINITION:​ if f(x) is continuous on [a,b], and differentiable on (a, b), there is some value “c” such that f(b)−f(a) ○ FORMULA​: f ′(C) = b−a   Arc Length of Parametrized Curves ● DEFINITION​: if C is a curve parametrized by x(t) = f(t) and y(t) = g(t) for a ≤ t ≤ b, the length can be estimated by partitioning C into n parts. 2 2 ○ FORMULA:​ ΔL 1 √ (f(1 )−f(t0)) +(g(t1)−g(t 0)   ○ This method uses the concept of Riemann sums so the smaller each partition (ΔL) , the more accurate the approximation of total arc length ○ Same concept using the Mean Value Theorem (MVT) 2 2 2 2 ■ FORMULA:​ ΔL 1 √ f ′t1) (b−a) +g (t′)1(b−a)   ● In General 5 n ○ L ≃ ∑ Δt f′(t ) +g′t )   k=1 √ k k b ○ FORMULA: ​L = ∫ f ′t) + g(′) dt a√ ● Note: Pay attention to any time you can find the arc length without using the formulas (using algebra or geometry instead) as it will save important time on tests. **Practice Problems on pg. 383 (#1, 2, and 4) and pg. 423 (#1, 2, and 8) Other Useful Information ● Pythagorean Theorem: a + b = c2 2 ● Trigonometric Identities ○ sin (x) +cos (x) = 1  ○ sin (x) = 1−cos(2x)  2 2 1+cos(2x) ○ cos (x) = 2   ○ sin(2x) = 2sin(x)cos(x)  ○ cos(2x) = (cos (x)−sin (x)) = (2cos (x)+1) = (1−2sin (x) ) ● Remember: ○ if the equation of a line is all under a square root, the line will only be above the x-axis ○ Make sure you are aware of when you need to take the derivative vs the integral ○ Check which function is on top/outside (just because it is labelled f(x) doesn’t mean it is in the position the formula refers to as “f(x)” 6


Buy Material

Are you sure you want to buy this material for

50 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

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!"

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 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."


"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.