Math 1B: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Summer1B/Many of the exercises are from Single Variable Calculus: Early Transcendentals for UC Berkeleyby James Stewart; these are marked with an §. Others are my own, are from the mathematicalfolklore, or are independently marked.Here’s a hint: drawing pictures — e.g. sketching graphs of functions — will always make theproblem easier.Applications of Integration — Physical GeometryStewart describes the following applications of integration to physics and engineering:• The pressure exerted by a fluid with density ρ at depth h is P = ρgh, where g = 10 m/s2isthe acceleration due to gravity. Thus, the total force on a flat plate is:F =Zbaρ(x)g x f(x) dxwhere ρ(x) is the density at depth x (for liquids under normal conditions ρ(x) = ρ is constant),and f(x) is the width of the plate at depth x (the plate lies between depths a and b, with0 ≤ 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 themasses, each mass “weighted” in the average by its mass (in fact, this is why such averageshave are called as such). For masses continuously distributed in one dimension, so that thelinear density at location x is ρ(x) (with ρ(x) = 0 outside the interval x ∈ [a, b]), the centerof mass is given by:¯x =Rbax ρ(x) dxRbaρ(x) dxThe denominator is the total mass m of the distribution. The moment of the distribution ism¯x.• If the masses are distributed in two or more dimensions, the moments and centers of massmay be computed dimension-by-dimension to give coordinates. The centroid of a region isthe center of mass of the region, where the region is given constant density. If a region witharea A is bounded by the curves y = f (x), y = g(x), x = a, and x = b, where a ≤ b andf(x) ≤ g(x) for x ∈ [a, b], then the centroid is located at (¯x, ¯y) where:¯x =1AZbax [g(x) − f(x)] dx ¯y =1AZba12(g(x))2− (f (x))2 dxOf course, A =Rba[g(x) − f (x)] dx.Here’s one more physics definitions:• The angular moment of inertia of a one-dimensional object with linear density ρ(x) (supportedon the interval [a, b]) is given byRbax2ρ(x) dx.Here are some math problems that use the above notions:11. § A vertical plate in the shape of an equilateral triangle with sidelength 2 m is submerged inwater (density ρ = 1 g/cm3) such that one edge is touching the surface of the water. Howmuch 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 surfaceof 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 dependsonly on the area of the plate (or rather of the part of the plate actually under the water) andon 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 shallowend having a depth of 3 ft and the deep end, 9 ft. If the pool is full of water, estimate thepressure on each of the five sides of the pool.5. Prove the following theorem of Archimedes: an object fully submerged in a fluid experiencesan upward “buoyant” force equal to the weight of the fluid that would fill the volume of theobject. Hint: consider first a normally-oriented rectangular box (you can prove the theoremfor boxes without calculus). Then use integral-style arguments to prove the theorem forarbitrary 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 = y2(c) § y = sin x, y = cos x, x = 0, x = π/4(d) y = ex, y = 0, x = 07. Find the moment and angular moment of a circle (with constant density ρ) of radius r locatedat (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? Theangular moment?9. Consider two distributions with equal total mass. Prove that the center of mass of thecombined 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 separatedistributions.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 thepiece of curve corresponding to the interval [x, x + dx] is dA = 2πf(x) ds, where ds =p1 + (f0(x))2dx. Find the total hydrostatic force felt by a sphere of radius 1 m submergedunder water so that the center is at a depth h (with h ≥ 1 m). Hint: orient the axes with xpointing down through the center of the sphere and x = 0 corresponding to the surface of
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