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_0^1 sec^2x e^(tan x)dx`
Here's the curve `y=sec^2x e^(tan x)`:
The shaded region represents the integral we need to find.
In the theory of lasers, we see
where `a`, `I_0` and `T` are constants. Find `E`.
Integrate each of the given functions.
`int x\ e^(-x^2)dx`
`int(4\ dx)/(sec\ x\ e\ ^(sin\ x)`
Here is the curve `y=1/e^(2-3x)`:
The shaded region represents the integral we just found.
Find the equation of the curve for which `(dy)/(dx)=sqrt(e^(x+3))` if the curve passes through `(1, 0)`.
The graph of the solution curve we just found, showing that it 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.)