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 TOPICS This page:   calculus    Search   Dr. Math See also the Internet Library:   calculus HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Calculus Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Chain rule.     Maximizing the volume of a box.     Maximizing the volume of a cylinder.     Volume of a tank.     What is a derivative? Volume of a Shape [11/11/1997] What is the volume of the shape formed by rotating the parabola y=x^2 around the line y=x? (From x = 0 to 1). Volume of a Solid [10/02/1997] The base of a solid is the region inside the circle x^2 + y^2 = 4... Volume of a Sphere [04/21/1998] Can you help me derive and prove the formula for the volume of a sphere? Volume of a Torus [04/22/1999] I would like to know how to find the volume of a torus using integrals. Volume of Intersecting Pipes [10/27/2000] How can you calculate the volume of the intersection of two perpendicular pipes of the same radius? Volume of Spherical Cap [02/06/1998] I am trying to find the volume of a cap of a sphere with radius of 5. The cap has a height of 3 - it is as if the top of the sphere, 3 meters from the top, was severed from the rest of the sphere. Volume Using Cross Sections [05/28/2001] Find the volume of the region between y = |x| and y = -|x|+6 with cross sections that are equilateral triangles and perpendicular to the x-axis. Washers and Discs, Graphs and Symmetry [2/20/1996] Use the washer method to find the volumes of the solid generated by revolving the regions bounded by the lines and curves about the y-axis... Water Draining from a Conical Reservoir [12/20/1995] A circular conical reservoir, vertex down, has depth 20 ft and radius of the top 10 ft. Water is leaking out so that the surface is falling at the rate of 1/2 ft/hr... Water in a Horizontal Tank [8/7/1996] What's the volume of water in a cylindrical tank 72" long and 36" in diameter, filled only to 4.25"? Water Rising in a Tank [3/7/1996] Water is rising in a tank with a sloping floor. Find how far it has risen after a certain amount of time. Water Trough Related Rate Problem [10/21/2003] A storage tank is 20 ft long and its ends are isosceles triangles having bases and altitudes of 3 ft. Water is poured into the tank at a rate of 4 (ft)^3/min. How fast is the water level rising when the water in the tank is 6 inches deep? Weierstrass Curve [09/13/2002] About 100 years ago the mathematician Weierstrass gave an example of a curve consisting of angles, or corners, and nothing else. Where can I find this equation? What Are Differential Equations? [12/17/2003] Can you please tell me what differential equations are? Why do they have two variables? Why are the solutions formulas and not numbers? What Does an Integral Represent? [07/09/2004] Use an integral to estimate the sum (from 1 to 10000) of sqrt(x). The definite integral is easy enough to formulate (i.e., 2/3*x^3/2, from 1 to 10000). But what does the integral represent? Surely it doesn't represent the sum of the square roots as required? What is 1^infinity? [12/10/1998] Why is 1^infinity indeterminate? How can you show this using limits? Why is it not equal to 1, as intuition leads us to believe? What is a continued fraction? [03/06/1998] What is a continued fraction and what makes it different from the types of fractions or ratios I'm used to? What is a Functional Transformation? [08/11/2002] I would like a brief definition of a functional transformation and what an application is in early Calculus. What is an Integral? [09/06/2001] I've never really understood integrals and how exactly they work. Could you please explain? What is dx? [08/25/2001] What does dx mean and where does it come from? What is Nonlinear Math? [10/02/1997] What exactly is nonlinear math, and what is it used for? What is the Integral of (tan x)^1/2 ? [03/07/1999] My first substitution was u^2 = tan x ... What is the Longest Pole? [04/04/1999] A metal pole L ft long is pushed on the floor from one corridor into another corridor at a right angle... When Did It Start Snowing? [02/17/1999] Convert to a differential equation: A snowblower throws 30 cu. ft. of snow per minute... When to Use the Chain and Quotient Rules [06/12/1998] In the following problem, would you use the chain rule or the quotient rule first to differentiate? Problem: (2x+1)^2/(2x+4). When to use the Shell Method or Washer Method to Calculate Volume [2/26/1996] We recently learned about the washer method and the shell method. I was wondering which method to use in different situations. Where Are Derivatives Used in Real Life? [03/20/2007] I've learned about derivatives in school, but I'm wondering where someone might apply them in real life? Where did e^(i*pi)+1 = 0 come from? [11/12/1997] Is there a way to solve the equation e^(i*pi)+1 = 0 without using or using very little calculus? Where is f Continuous? [10/07/1997] Graph the following... Besides at x = 3, where is f continuous? Where to Land on the Beach? [5/22/1996] An island is 6 miles offshore... where should you land in order to go from the island to a store on shore in the least possible time? Why Are 1^infinity, infinity^0, and 0^0 Indeterminate Forms? [05/08/1998] Using limits to prove that 1^infinity, infinity^0, and 0^0 are indeterminate forms. Why Are the Rates of the Sliding Ladder Problem Different? [12/17/2003] In the classic related rates problem where a ladder slides down a wall, why are the rates of the top and bottom of the ladder different? It seems to me that if you pull the bottom at a constant rate, the top should also slide down the wall at the same constant rate. Why Can't Some Functions be Integrated? [11/06/1996] How do I evaluate Integral[x tanx dx]? Why Differentiability Implies Continuity [03/06/1998] Why must a function that is differentiable in an interval be continuous in that interval? Why Does Height Formula Use -16 Instead of -32? [01/31/2008] If the acceleration of gravity is -32 ft/sec^2, why does the formula for the position of a falling object (h = -16t^2 + s) use -16 instead? Why Does Integration by Substitution Work? [12/13/2006] I just learned integration by substitution. Part of my textbook's explanation seems to depend on treating differentials as numbers that cancel out. But I've also seen that differentials cannot be treated that way. Can you clarify that and explain why substitution works? Why Is dx Necessary? [01/31/2002] In integral notation where you have one variable after an integral sign you must always end with the change in your variable (i.e.: INT[] x^2 dx). Why is the 'dx' necessary? Why Should I Study Calculus? [03/04/2001] I am starting an AP calculus class next year and I would like to know why I should study calculus. Why use the natural log...? [1/9/1995] Why do we have to use the natural log for the antiderivative of 1/x? Why doesn't the power rule work on the antiderivative of 1/x? Work Needed to Lift a Leaky Bucket [08/01/1999] How can you calculate the amount of work needed to lift a bucket of water to the top of a well if there is a hole in the bucket causing water to leak out at a rate proportional to the height of the water in the bucket? Page: []

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