Volume - Solid Revolution

Category: Integral Calculus, Analytic Geometry

"Published in Newark, California, USA"

Find the volume generated by revolving about the x-axis and y-axis the areas bounded by the curves

Solution:

To illustrate the problem, it is better to sketch the graph of the three equations above using the principles of Analytic Geometry as follows

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From the figure above, the point of intersection between y = x³ and y = 0 is (0, 0), the point of intersection between y = 0 and x = 2 is (2, 0), and the point of intersection between y = x³ and x = 2 is (2, 8) by substituting x = 2 to y = x³. 

Next, from the given three equations above, it is better to use a vertical strip at the area bounded by three curves and label further the figure as follows

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If you rotate the shaded area about the x-axis, the vertical strip becomes a disk as follows

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The volume of a disk formed by the rotation of a vertical strip about the x-axis is 

Integrate on both sides of the equation to get the volume of a solid formed by the rotation of the area about the x-axis as follows

If you rotate the shaded area about the y-axis, the vertical strip becomes a cylindrical shell as follows

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The volume of a cylindrical shell formed by the rotation of a vertical strip about the y-axis is 

The circumference of the base of the cylindrical shell is 2πx. x is the distance of the vertical strip to the axis of revolution which is the y-axis. y is the height of a cylindrical shell. dx is the thickness of a cylindrical shell. Since dx is a very small value, then the two radii of the cylindrical shell are almost the same which is x. The three dimensions of a thin rectangular box are 2πx, y, and dx. When you wrapped a thin rectangular box into a cylinder, then it becomes a cylindrical shell. Therefore, the volume of a cylindrical shell is

Integrate on both sides of the equation to get the volume of a solid formed by the rotation of the area about the y-axis as follows

Word Problem - Exponential Decay

Category: Chemical Engineering Math, Differential Equations, Integral Calculus

"Published in Newark, California, USA"

Radium decomposes at a rate proportional to the quantity of radium present. Suppose that it is found that in 25 years approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half the original amount of radium to decompose.

Solution:

The given word problem is about the decomposition of a substance in a certain period of time. This is exactly the opposite of exponential growth as we discussed in the previous topics. If the statement says "Radium decomposes at a rate proportional to the quantity of radium present.", then the working equation will be

or

The sign for constant of proportionality is negative since it is a decomposition process or an exponential decay. If it is an exponential growth, then the sign is positive.

Let

^dx/_dt = be the rate of decomposition of a radium 
     x = be the amount of radium at time t 
   x₀ = be the initial amount of radium at time t = 0

Consider the above equation

Solve the above equation using the Separation of Variables, as follows

Integrate both sides of the equation, we have

Take the inverse natural logarithm on both sides of the equation

To solve for the value of C, we need the following condition: If x = x₀ at t = 0, then the above equation becomes

Substitute the value of C to the above equation, we have

Next, we need to solve for the value of k which is the constant of proportionality. If the next statement says "Suppose that it is found that in 25 years approximately 1.1% of a certain quantity of radium has decomposed.", then the following condition will be as follows

Let

x = (1 - 0.011)x₀ = 0.989x₀
t = 25 years

Substitute the values of x and t to the above equation, we have

Take the natural logarithm on both sides of the equation

The final working equation will be

If x = 0.5x₀, then radium will decompose in

Take natural logarithm on both sides of the equation