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# Math 141: Exam 1 Study Guide 141

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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.

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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 = tso 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

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