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Solving the integral yields:
2ln
2
1
2ln
2
1
2ln02ln2ln
1ln
2
2
ln0cosln
4
coslncoslntan
4
0
4
0
yydy
So the first quadrant area bounded by the following curves:
,
and
is
equal to
square units.
3. Area between two curves.
This can be considered as a more general approach to finding areas. Thus each of the previous
examples could have been solved using such an approach by considering the x- and y- axes as
functions with equations y=0 and x=0, respectively.
Many areas can be viewed as being bounded by two or more curves. When area is enclosed by just
two curves, it can be calculated using vertical elements by subtracting the lower function from the
upper function and evaluating the integral.
Analogously, to calculate the area between two curves using horizontal elements, subtract the left
function from the right function.
As always, a sketch of the graph can be a very important tool in determining the precise set-up of
the integral. If you subtract in the wrong order, your result will be negative. That mistake can be
avoided by taking the absolute value of the difference of the functions.
Here is the universal formula for finding the area between two curves:
Using the vertical elements:
where y
1
and y
2
are functions of x
Using the horizontal elements:
where x
1
and x
2
are functions of y.
Ex.5. Find the area of the region enclosed by the following curves:
,
,
and
.
As always, we will first draw a sketch.