Integrals

[stillofthenight]

A function f(x)=e^-Ax. A is a positive real number.

Show that the integral from 1 to 2 of f(x) dx ---> 0 as A ---> infinity.

From plotting e^-x I get a reflection across the y axis of e^x. As A approaches infinity I get a positive sloped line through the origin and it becomes less defined of a slope as A approaches infinity which means the area of the integral becomes closer to zero.

Does this make any sense of what I am saying , is this correct, am I showing it? How else can you verify it?

 

[daon2]

Integrate it and calculate the actual limit.

 

[DrPhil]

A function f(x)=e^-Ax. A is a positive real number.

Show that the integral from 1 to 2 of f(x) dx ---> 0 as A ---> infinity.

From plotting e^-x I get a reflection across the y axis of e^x. As A approaches infinity I get a positive sloped line through the origin and it becomes less defined of a slope as A approaches infinity which means the area of the integral becomes closer to zero.

Does this make any sense of what I am saying , is this correct, am I showing it? How else can you verify it?

The function e^x does NOT reflect across the y-axis. Your sketch should not go "through the origin." Instead, the value is
f(0) = e^0 = 1, independent of A.
For all x>0, the function is continuously decreasing, and the slope is also continuously decreasing [the curve gets smaller and flatter]. The larger A is, the sharper the initial drop, but it would be hard to justify the integral going to zero unless you do as daon2 said:

Integrate it and calculate the actual limit.