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:

math expression

It is remarkable because the integral is the same as the expression we started with: eu.

Example 1: math expression


Answer

Example 2: math expression


Answer

Example 3: math expression


Here is the LiveMath solution to this problem.

LIVEMath


Answer

Example 4:

In the theory of lasers, we see

math expression

where a, I0 and T are constants. Find E.


Answer

 

Exercises

Integrate each of the given functions.

1. math expression


Answer

2. math expression


Answer

3. math expression


Answer

4. Find the equation of the curve for which math expression if the curve passes through (1,0).


Answer

 

Application - Volume of Solid of Revolution

The area bounded by the curve y = ex, 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.)


Answer



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