Using Shell or Disc Method to Find Volume of the Solid

Disc method and Shell(cylinder) method of integration are the two different methods of finding volume of solid of a revolution, using rectangular coordination system the functions are defined in terms of x in the below problem.

Topic : Disc or Cylinder Method of Finding Volume of the Sphere.
Problem : Use the disc or shell method to find the volume of the solid generated by revolving the regions bounded by the graphs of the equations about the y axis. y=x³, y=0, x=2
Solution :

Volume of a Solid by rotating about y-axis is given by:

V = 2π_a∫^bp(x)h(x) dx
here p(x)=x³, h(x)=x
when y = 0, 0 = x³ or ³√0 = x or x = 0
So a = 0 and b = 2
Plugging in all the values in the formula, we get

V = 2π₀∫²(x)³.x dx

= 2π₀∫⁸(x)⁴ dx

= 2π[x⁵/5₀]²

= 2π[(2)⁵/5 - (0)⁵/5]

= 2π[32/5]

= 64π/5

So this how the volume of Solid of revolution is determined when the equations about the y axis.
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Problem on Logarithm and Solving for the value of x

Logarithmic Expression are solved using Logarithmic rules/identities and find unknown value of a variable x.

Topic : Solving for x using Logarithm

Below problem explains use of logarithmic identities/properties and also quadratic equation.

Problem : log 3 (x + 11) + log 3(x + 5) = 3, Solve for x

Solution :

We have the property of log as

log a + log b = log (ab)

So, log 3 (x + 11) + log 3(x + 5) = log (3(x + 11)3(x + 5)) = log 9(x² + 16x + 55)

Thus we have ; log 9(x² + 16x + 55) = 3

as we have the base of given logarithmic expression 10,

So 9(x² + 16x + 55) = 10³ (by the definition of logarithm identity)

9x² + 144x + 495 = 1000

9x² + 144x + 495 - 1000 = 0

9x² + 144x - 505 = 0

x = -144 ±√[144² - 4 (9)(-505)] / 2(9)

= -144 ±√[20736 + 18180] / 18

= -144 ±√[38916] / 18

= -144 ± 197.27 / 18

Thus either x = 2.96 or -18.96

But as log of a negative number is not defined, we discard x = -18.96

hence the answer is x = 2.96


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