Logarithmic Functions and Trigonometry in Calculus

Before calculus was discovered, formulas such as that for the volume of a sphere, were figured out using methods of exhaustion. Archimedes in particular contributed greatly in this regard.

This post will introduce some of the basic concepts of integration and provide a detailed example for determining the volume of a sphere.

As previously discussed trigonometry and natural logarithmic functions play an important role in Calculus. I’ll begin first with an introduction to logarithmic functions in Calculus. In future posts I’ll discuss in greater detail logarithmic functions and provide some more examples.

In differential calculus the general derivative of a logarithmic function is as follows.

     \[   f(x) = log _b x  \] \[   f'(x) = \frac{1}{x ln(b)}  \]

For examples of greater complexity we implement the use of the chain rule but for reasons of simplicity consider the following example of the differentiation of natural logarithmic functions.

     \[   f(x) = ln x  \] \[   f'(x) = \frac{1}{x}  \]

I mentioned before that integration and differentiation are inverses of one another. This concept is of great importance as we have seen by the use of anti-derivatives to solve integrals. By this logic the following should be clear.

     \[   f(x) = \frac{1}{x}   \] \[   F(x) = \int{\frac{1}{x}}\,\mathrm{d}x   \] \[   F(x) = ln x + C   \]

An interesting example of the inverse relationship between differentiation and integration can be seen in the following special case of the exponentiation of the natural e.

     \[   f(x) = e ^ x   \] \[   f'(x) = e ^ x   \]

It can be proven using the “Exponentiation Rule” that the differential of the function e to the power of x is simply e to the power of x. By this logic the following should also be apparent.

     \[   f(x) = e ^ x   \] \[   F(x) = \int{e ^ x}\,\mathrm{d}x   \] \[  F(x) = e ^ x + C  \]

In a past post regarding the area under the normal curve I demonstrated the use of Exponentiation Rule and anti-derivatives for the function of the natural power of e.

Furthermore, I will now discuss the importance of trigonometry in Calculus. The use of trigometry in calculus was also demonstrated in the example where I found the area under the normal curve. In this example it was fundamental in the conversion of cartesian to polar coordinates.

I wont discuss the basics of trigonmetric differentiation or integration in this post but I will however provide an extensive example of how its used. The example I will show you is how to calculate the volume of a sphere by spinning a circle about the y axis and integrating.

We begin by determining that the area of a cirle is pi times its radius squared. To accomplish this we make use of the general equation for a circle.

      \begin{align*}      f(x) & = \sqrt{r ^ 2 - x ^ 2 } \\   A &= 2 \int_{-r}^r{\sqrt{r^2 - x^2}} \,\mathrm{d}x \\   &=  4 \int_{0}^r{\sqrt{r^2 - x^2}} \,\mathrm{d}x \\     & =  4 \int_{0}^{\frac{\pi}{2}}{\sqrt{r^2 - (r^2 Cos^2\theta)} r Sin \theta} \,\mathrm{d}\theta \\  & =  4 \int_{0}^{\frac{\pi}{2}}{\sqrt{r^2  (1 - Cos^2\theta)} r Sin \theta }\,\mathrm{d}\theta \\  & =  4 \int_{0}^{\frac{\pi}{2}}{\sqrt{r^2  (Sin^2 \theta  ) } r Sin \theta } \,\mathrm{d}\theta \\   &=  4 \int_{0}^{\frac{\pi}{2}}{r  Sin \theta r Sin \theta } \,\mathrm{d}\theta \\   &=  4 r^2 \int_{0}^{\frac{\pi}{2}}{  Sin ^2 \theta } \,\mathrm{d}\theta \\   &=  4 r^2 \int_{0}^{\frac{\pi}{2}}{ \frac{1}{2} \left( 1 - Cos 2 \theta \right)  } \,\mathrm{d}\theta \\  & =  4 r^2 \int_{0}^{\frac{\pi}{2}}{ \left[ \frac{1}{2} - \frac{Cos 2 \theta}{2} \right] } \,\mathrm{d}\theta \\   &=   2 r^2 \int_{0}^{\frac{\pi}{2}}{  } \,\mathrm{d}\theta  -  2 r^2 \int_{0}^{\frac{\pi}{2}}{ Cos 2 \theta } \,\mathrm{d}\theta \\  & =   2 r^2 \left[ \theta \right]_0^{\frac{\pi}{2}} - 2  r^2 \left[ \frac{Sin 2 \theta}{2} \right]_0^{\frac{\pi}{2}} \\  & =   2 r^2 \left[ (\frac{\pi}{2}) - (0) \right] - 2  r^2 \left[ (0) - (0) \right] \\  & =  \pi r ^ 2\\   \end{align*}

Now we can calculate the volume of the sphere by determining the sum of the infintesimal disks along the x-axis. Therefore we can use the following simple integral.

     \begin{align*} V &=  \int_{-r}^r{\pi y^2}\,\mathrm{d}x  \\ V &=  2 \int_{0}^r{\pi y^2}\,\mathrm{d}x  \\ V &=  2 \pi \int_{0}^r{( \sqrt{r ^ 2 - x ^ 2 } )^2}\,\mathrm{d}x  \\ V &=  2 \pi \int_{0}^r{r ^ 2 - x ^ 2 }\,\mathrm{d}x  \\ V &= 2 \pi \int_{0}^r{r ^ 2}\,\mathrm{d}x  -  2 \pi \int_{0}^r{x ^ 2 }\,\mathrm{d}x  \\ V &=  2 \pi r ^ 2 \int_{}^r \,\mathrm{d}x  - 2  \pi \int_{}^r{x ^ 2 }\,\mathrm{d}x  \\ V &=  2 \pi r ^ 2  \left[  x \right] _{0}^r  - 2 \pi \left[ \frac{x ^ 3}{3} \right] _{0}^r  \\ V &=  2 \pi r ^ 2  \left[  (r) - (0) \right] - 2 \pi \left[  \left( \frac{(r) ^ 3}{3} \right) -   (0)  \right]  \\ V &=  2 \pi r ^ 3   -  \frac{2 \pi r ^ 3}{3}  \\ V &=  \frac{6 \pi r ^ 3}{3}   -  \frac{2 \pi r ^ 3}{3}  \\ V &=  \frac{4 \pi r ^ 3}{3}  \\ \end{align*}

So as you can see integral calculus can be a powerful tool in both science and mathematics.

I will show a few more examples of interesting problems such as those done by Archimedes in my future posts.

Dan Roy, Founder and Creator of Universal Physics.

Category(s): Mathematics
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140 Responses to Logarithmic Functions and Trigonometry in Calculus

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