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Calculus and Analysis

Surface of the area of revolution

We are supposed to use the definition of the Area of a Surface of Revolution to solve this problem. I have attached this formula and the answers I received in a word document. The problem: Given: y = -x^2 + 4x defined on the closed domain [0,4] Revolve the graph about the x-axis. Find the area of the surface obtained

Calculate: Centroid of a Plane Region

I need to see how to find the centroid coordinates by using integrals and moments. I have attached a word document with the formulas we are supposed to use to find the centroid. Now here is the problem: Given: y = 9 - x^2, y = 2 Find the coordinates of the centroid of the above plane region. Please refer to the atta

Shell method to find volume

Given: f(x)=2x, g(x)=10 a)Sketch the plane region bounded by the functions graphs and the y-axis b)Use the shell method to find the volume of the solid formed by revolving the above plane region about the y-axis. NOTE: the graph I did is attached. The answer I got was 523.599. I'm trying to check to see if I did the gr

Use the Washer Method to find volume

Given: f(x)=2x, g(x)=x, x=5 a)Sketch the plane region bounded by the functions graphs b)Use the washer method to find the volume of the solid formed by revolving the above plane region about the x-axis.

Solving differential equations

Solve each of the following differential equations: ***For each problem,state the method you used and show the work required to obtain the answer.*** 1) (y-(cos^2)x)dx + cosxdy=0 2) ye^x dx= (4+e^2x)dy

Functions : Critical Values

The function f(x)=2x^3 - 33x^2 + 108x - 6 has two critical numbers. The smaller one equals ______ and the larger one equals______.

Differential equation from calculus II

Given the differential equation: (y^4)(e^2x) + y' = 0 NOTE: The differential equation above is attached in a microsoft word document for better legibility. Additionally my work is attached as a jpeg file. The questions: a)Find the general solution. b)Find the particular solution such that y(0) = 1.

Newton's Law of Cooling relating to differential equations.

At 10:00 AM, an object is removed from a furnace and placed in an environment with a constant temperature of 68 degrees. Its core temperature is 1600 degrees. At 11:00 AM, its core temperature is 1090 degrees. Find its core temperature at 5:00 PM on the same day.

Increasing functions: Comparing Functions

Explain why the graph of f(x) is rising over an interval a < or equal to x < or equal to b if f '(x) > 0 throughout the interval. What can you say about the graph of f if f '(x) is less than zero on a < or equal to x < or equal to b?

Sample Question: Word problems

1. The manager of a large apartment complex knows from experience that 80 units will be occupied if the rent is 320 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decreas


1. A weather balloon is rising vertically at a constant rate of 4 ft/s directly above a straight and level road. When the balloon is 75 ft above the road, a car moving at 55 ft/s passes directly under the balloon. Based on this information find: a. the rate the distance between the balloon and the car is changing 3 sec after t

A minimization fencing problem.

A rectangular field is going to be enclosed and divided into two separate rectangular areas. (Areas do not have to be equal). Find the minimum fencing that is required if the total area of the field is 1200m2.

Working with infinite sequences and series.

If series Sum(an) and Sum(bn) with positive terms are convergent, is the series Sum(an*bn) converegent? Note: 1. Sum replaces the symbol for summation 2. an and bn are nth elements of the two series

Calculating rates of change in a loan situation.

The formula for the loan one can get with a payment of $P paying monthly for 15 years at an interest rate of r is: L=(12P/r)[1-(1+(r/12))^(-180)] a.) Find dL/dt, the rate of change of the loan with respect to time. (Here, t is the time that is passing, not the t in the original function if you know the loan. Trea

Exponential growth and decay

A leaking oil tank has a capacity of 500 000 liters of oil. The rate of leakage depends on the pressure of oil remaining in the tank and the pressure depends on the height of oil. When the tank is half-full, it loses 20L/min. How long goes it take to lose 15 000L from half-full?

Explaination for derivatives

Explaination for derivatives related to exponential and logarithmic functions ,formulae used to solve them and solutions to some problems. All problems are in the solution file

What is the equation of three bisecting solid rods centered at the origin?

Hello, What is the equation of three bisecting solid rods centered at the origin? Given 3 solid rods of length 3 and diameter 1. One rod is on the x-axis One rod is on the y-axis and One rod is on the z-axis Each is centered at the origin and is perpendicular to the other rods in each axis. Need equation in rectan

When to use the Chain Rule

The key is whether or not you are plugging the result of a function into another function. The idea is shown by contrasting the procedures for taking the derivatives of sin(x^2) and x^2*sin(x).

Euclidean space

Compute the distance from a point b = (1, 0, 0, 1)^T to a line which passes through two points (0, 1, 1, 0)^T and (0, 1, 0, 2)^T. Here ^T denotes the operation of transposition, i.e. the points are represented by column-vectors instead of row-vectors.