Area
Application of definite integral for determination of
areas (simple cases).
Basic Concepts
Let y = f1 (x) and y = f2 (x) be
two given curves, which are continuous in [a, b]. Suppose we have to find
the area of the plain region bounded by the two curves .
y = f1(x) and y = f2(x) between
the ordinates x = a and x = b (i.e. the area of the region A1B1C1D1). .
For this purpose we
take an elemental strip ABCD of width dx and height AD (or BC). .
Clearly AD @ BC = f1(x) ~ f2(x).
Now the area of this
strip,
dA = (f1(x)~f2(x))
dx
Thus the required
area,
A=
![](http://www.quizsolver.com/radix/dth/notif/Area_files/image002.gif)
Remarks :
- If f1(x) ≥ f2(x) " x ∈ [ a, b] then A =
![](http://www.quizsolver.com/radix/dth/notif/Area_files/image003.gif)
- If f1(x) £ f2(x) " x ∈ [ a, b] then A =
![](http://www.quizsolver.com/radix/dth/notif/Area_files/image004.gif)
- If f1(x)
≥ f2(x) " x ∈ [ a, c]
- and f1(x)
£ f2(x) " x ∈ [ c, b],
- then =
![](http://www.quizsolver.com/radix/dth/notif/Area_files/image006.gif)
- We can finally
conclude that the area bounded by the two curves between
x = a and x = b is ![](http://www.quizsolver.com/radix/dth/notif/Area_files/image007.gif)
- If f2(x)
= 0 " x ∈ R,
would give us area
bounded by y = f1(x) and the x-axis between the ordinates x = a
and x = b (here we are assuming that f1(x) ≥ 0 " x ∈
[ a, b] ).
Similarly
would give us the area bounded by the
curves x = f1(y) and x = f2(y) between the lines y =
c and y = d
![](http://www.quizsolver.com/radix/dth/notif/Area_files/image010.gif)