Math 1B Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Summer1B Many of the exercises are from Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart these are marked with an Others are my own are from the mathematical folklore or are independently marked Here s a hint drawing pictures e g sketching graphs of functions will always make the problem easier Applications of Integration Physical Geometry Stewart describes the following applications of integration to physics and engineering The pressure exerted by a fluid with density at depth h is P gh where g 10 m s2 is the acceleration due to gravity Thus the total force on a flat plate is Z b F x g x f x dx a where x is the density at depth x for liquids under normal conditions x is constant and f x is the width of the plate at depth x the plate lies between depths a and b with 0 a b we measure depths down from the surface The center of mass of a distribution of masses is a weighted average of the locations of the masses each mass weighted in the average by its mass in fact this is why such averages have are called as such For masses continuously distributed in one dimension so that the linear density at location x is x with x 0 outside the interval x a b the center of mass is given by Rb x x dx x Ra b a x dx The denominator is the total mass m of the distribution The moment of the distribution is mx If the masses are distributed in two or more dimensions the moments and centers of mass may be computed dimension by dimension to give coordinates The centroid of a region is the center of mass of the region where the region is given constant density If a region with area A is bounded by the curves y f x y g x x a and x b where a b and f x g x for x a b then the centroid is located at x y where 1 x A Of course A b Z 1 y A x g x f x dx a Rb a Z a b 1 g x 2 f x 2 dx 2 g x f x dx Here s one more physics definitions The angular moment of inertia of a one dimensional object with linear density x supported Rb on the interval a b is given by a x2 x dx Here are some math problems that use the above notions 1 1 A vertical plate in the shape of an equilateral triangle with sidelength 2 m is submerged in water density 1 g cm3 such that one edge is touching the surface of the water How much pressure is applied to one side of the plate Be careful with units 2 If the plate in the previous problem is rotated 180 so that its upper point touches the surface of the water what is the total pressure applied to one side of the plate 3 Prove that the pressure applied to one side of a plate submerged vertically in water depends only on the area of the plate or rather of the part of the plate actually under the water and on the depth of its centroid 4 A swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined plane the shallow end having a depth of 3 ft and the deep end 9 ft If the pool is full of water estimate the pressure on each of the five sides of the pool 5 Prove the following theorem of Archimedes an object fully submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid that would fill the volume of the object Hint consider first a normally oriented rectangular box you can prove the theorem for boxes without calculus Then use integral style arguments to prove the theorem for arbitrary objects 6 Sketch the region bounded by the given curves and find the centroid a y ex y 0 x 0 x 1 b y x2 x y 2 c y sin x y cos x x 0 x 4 d y ex y 0 x 0 7 Find the moment and angular moment of a circle with constant density of radius r located at a 0 8 Consider a one dimensional distribution with mass m moment M and angular moment I If you move the object one unit to the right what happens to the mass The moment The angular moment 9 Consider two distributions with equal total mass Prove that the center of mass of the combined distribution is halfway between the centers of masses of the separate distributions What happens when the distributions have different masses 10 Prove that the moment of a sum of two distributions is the sum of the moments of the separate distributions 11 Prove that the centroid of any triangle is located at the point of intersection of the medians 12 Recall that when the curve y f x is rotated around the x axis the surface area of the piece of curve corresponding to the interval x x dx is dA 2 f x ds where ds p 1 f 0 x 2 dx Find the total hydrostatic force felt by a sphere of radius 1 m submerged under water so that the center is at a depth h with h 1 m Hint orient the axes with x pointing down through the center of the sphere and x 0 corresponding to the surface of the water 2
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