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Selected answers to common questions:
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
- 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?