3. Integration: The Exponential Form
by M. Bourne
By reversing the process in obtaining the derivative of the exponential function, we obtain the remarkable result:
It is remarkable because the integral is the same as the expression we started with. That is, `e^u`.
`int_0^1sec^2x\ e^(tan\ x)dx`
In the theory of lasers, we see
where `a`, `I_0` and `T` are constants. Find `E`.
Integrate each of the given functions.
1. `int x\ e^(-x^2)dx`
2. `int(4\ dx)/(sec\ x\ e\ ^(sin\ x)`
4. Find the equation of the curve for which `(dy)/(dx)=sqrt(e^(x+3))` if the curve passes through `(1, 0)`.
Application - Volume of Solid of Revolution
The area bounded by the curve `y = e^x`, the `x`-axis and the limits of `x = 0` and `x = 3` is rotated about the `x`-axis. Find the volume of the solid formed. (You may wish to remind yourself of the volume of solid of revolution formula.)
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