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Browse College Calculus

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Testing a Point on a Sphere [01/14/1997]
What mathematical equation do you use to define a curved line segment on the surface of a sphere?

Tipped and Partially Filled Frustum [12/14/2003]
A vessel in a plant where I work is the frustum of a cone on its side. A liquid is contained in this section and pours out the end of the cone section, therefore the liquid only takes up a certain portion of the cone's volume. How can I compute the volume of the liquid?

Tips on Integrating Trig Functions [02/01/2001]
How can I integrate trig functions like dx/(3+4sin x)^2 and ((tanx)^2+2)^0.5? Do I have to get a reduction formula to integrate (1- x^2)^n from (0,1), where n is an integer?

Tips on Solving Differential Equations [10/27/2003]
I have reached a total impasse with the following questions: xdy = (x^5 + x^3*y^2 + y)dx (y + x)dy = (y - x)dx xdy = (y + x^2 + 9y^2)dx

Triple Integration [05/09/2003]
Consider a frustum with a lower circular base radius equal to R, and an upper circular base radius and the height both equal to R/2. If the center of the lower base is at the origin, evaluate the triple integral...

Trying to Integrate f(x) = exp(-ax^2) [11/23/2001]
How would you integrate a function of the form f(x) = exp(-ax^2)?

Tuning a Piano [05/28/2002]
Each piano has a unique tuning curve that can be approximated through 3-point quadratic interpolation. Is there a formula that will allow me to measure three different notes on the piano and find the tuning curve in between these notes?

Two Integration Problems [05/26/1999]
How can I integrate x[sqrt(1-x^2)]? How can I find the volume of the torus formed by rotating the circle x^2 + (y - R)^2 = r^2 [r <= R] about the X-axis?

A Two-Variable Delta-Epsilon Proof [02/05/2001]
Can you show me a delta-epsilon proof that 0 is the limit of (x*sqrt(1+y) + y*sqrt(1+x)) as (x,y) goes to 0?

Unit and Basis Vectors in Three Dimensions [05/09/1998]
Explanations and uses of unit vectors and basis vectors.

Using Newton's Method to Solve an Implicit Equation [10/14/2007]
How can I solve the equation (sin(X))/X = 0.7031? My book gives the answer but does not explain how to find it.

Variation of Parameters [11/10/2000]
How can I solve the differential equation y" + 4y = sec(x) using variation of parameters?

Vector Calculus [2/8/1996]
Find the total distance travelled by a particle along the path, sketch the path; find the equation of the tangent line at t1, sketch the curve and tangent at t1...

Vector Calculus [02/26/1998]
The vector field, F = xj, is given. Show that the line integral of F around any closed curve in the xy-plane, oriented as in Green's Theorem, is equal to the area of the region enclosed by the curve.

Volume of a Conical Wedge [02/19/2004]
What is the formula for the volume of the shape that remains when a cone is sliced by a plane parallel to the height of the cone, and the chord created by the slice on the base circle is less than the diameter?

Volume of an Elliptical Cone [10/15/1998]
Can you help me on find the volume for an elliptical cone by using a triple integration?

Volume of a Paraboloid Section [05/19/2000]
How do you evaluate the triple integral of (z dx dy dz), where omega is bounded from below by z=x^2+y^2, and bounded from above by z=10-2x-y? I can't find the limits or the boundary.

Volume of a Partially Filled Cylindrical Tank [04/28/2000]
I have a cylindrical tank on its side, and want to mark a dipstick for it.

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 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.

Walking Around the World [02/08/1997]
Given a starting point on the earth and a direction, if a man travels a certain distance in a straight line, where will he end up and in which direction will he be facing?

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 Are Elliptic Curves and Why Are They Useful? [03/04/2006]
What are elliptic curves? Why are they important?

What are Lagrange Multipliers? [02/09/2000]
Can you explain Lagrange multipliers and show me how they can be used to find the dimensions of a box which minimizes its cost?

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.

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 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?

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