Home > Calculus > Integration: the Area Problem

What is the net area between #f(x) = 2/(x+3) # and the x-axis over #x in [1, 2 ]#?

Answer 1

#ln(25/16)#

The area between the #x# axis and our function can be given by the integral of the function #f(x)# with the appropriate limits applied to the integral:

#int_1^2f(x)dx=int_1^2 2/(x+3)dx#

We can read the integral from a table of standard integrals to obtain:

#=[2ln|x+3|]_1^2#

Now applying the limits:

#={2ln|2+3|}-{2ln|1+3|}#

#=2(ln5-ln4)#

Applying our rules of logarithms to do a bit of tidying up and we get:

#=ln(25/16)#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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