`int_0^2 e^(-x) dx` Decide whether the integral is improper. Explain your reasoning

Expert Answers

An illustration of the letter 'A' in a speech bubbles

The integral does not have infinite bounds and the function is well defined over the whole interval of integration so there is no need to use limits. Therefore, the integral is not improper.

`int_0^2 e^-x dx=`

Substitute `u=-x` `=>` `du=-dx,` `u_l=0,` `u_u=-2.`

`u_l` and `u_u` denote lower and upper bound of integration.

...

See
This Answer Now

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Get 48 Hours Free Access

The integral does not have infinite bounds and the function is well defined over the whole interval of integration so there is no need to use limits. Therefore, the integral is not improper.

`int_0^2 e^-x dx=`

Substitute `u=-x` `=>` `du=-dx,` `u_l=0,` `u_u=-2.`

`u_l` and `u_u` denote lower and upper bound of integration.

`-int_0^-2 e^u du=int_-2^0 e^u du=e^u|_-2^0=e^0-e^(-2)=1-e^-2`

As we can see there was no need to use limits for calculating the integral.

The image below shows the graph of the function. We can see from the image that the function is not only defined but continuous over the whole interval of integration. In fact, domain of exponential function is set of all real numbers (can be extended to all complex numbers).

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)
Approved by eNotes Editorial Team