1.5 Area Between Curves
There are two types of areas between curves:
Type(I) Type(I) x y(y)
=
fe S"
f(x)
-n
=
- g(x)
=
=
x ax b
=
Type I:
xx*
->
y g(x)
=
f(xx)
= ~
g(Xk*)
=
x ax b
=
X1 &k
before, divide equal parts of width the kit sub interval
as
[Xx 1,Xx]
Ca, b1
into n AX=
9 on
-
Choose X,* and estimate the area of the strip by the area of the rectangle height
of
f(x,*) -
g(Xx*)
Ak [f(xx*)
=
-
g(Xx*))AK
add all areas and take the limitas n =
x
A
=
4..; (f(x*)-g(x,*)IAX Sa(fIX) -g(x)Idx
=
limit of a Reimann sum
SteDS:
1) Sketch the graphs and identify the region
2) draw a typical'rectangle
3)constructthe integral (s)
eg. Ofind the area bounded by y ex and
y
x2-1 on [-1,27
=
=
exy xa-
I
=
i- typical scie
A S_[ex- (x2-1)]dx
=
:J.Sex x +11dx -
2
ex
x x
=
- +
10 +
=2
2 1 1)
+
-
2
+ - -
1
=er--
There are two types of areas between curves:
Type(I) Type(I) x y(y)
=
fe S"
f(x)
-n
=
- g(x)
=
=
x ax b
=
Type I:
xx*
->
y g(x)
=
f(xx)
= ~
g(Xk*)
=
x ax b
=
X1 &k
before, divide equal parts of width the kit sub interval
as
[Xx 1,Xx]
Ca, b1
into n AX=
9 on
-
Choose X,* and estimate the area of the strip by the area of the rectangle height
of
f(x,*) -
g(Xx*)
Ak [f(xx*)
=
-
g(Xx*))AK
add all areas and take the limitas n =
x
A
=
4..; (f(x*)-g(x,*)IAX Sa(fIX) -g(x)Idx
=
limit of a Reimann sum
SteDS:
1) Sketch the graphs and identify the region
2) draw a typical'rectangle
3)constructthe integral (s)
eg. Ofind the area bounded by y ex and
y
x2-1 on [-1,27
=
=
exy xa-
I
=
i- typical scie
A S_[ex- (x2-1)]dx
=
:J.Sex x +11dx -
2
ex
x x
=
- +
10 +
=2
2 1 1)
+
-
2
+ - -
1
=er--
