Analisi matematica/Esempi di calcolo di integrali definiti
∫
1
a
1
x
d
x
=
(
log
x
)
1
a
=
log
a
{\displaystyle \qquad \int _{1}^{a}{1 \over x}dx=(\log x)_{1}^{a}=\log a}
∫
γ
(
x
+
y
)
d
x
{\displaystyle \qquad \int _{\gamma }^{}(x+y)dx}
essendo
γ
{\displaystyle \ \gamma }
y
=
a
x
2
{\displaystyle \ y=ax^{2}}
x
=
0
{\displaystyle \ x=0}
x
=
m
:
{\displaystyle \ x=m:}
∫
γ
(
x
+
y
)
d
x
=
∫
0
m
(
x
+
a
x
2
)
d
x
=
(
x
2
2
+
a
x
3
3
)
0
m
=
m
2
2
+
a
m
3
3
{\displaystyle \int _{\gamma }^{}(x+y)dx=\int _{0}^{m}(x+ax^{2})dx=({x^{2} \over 2}+a{x^{3} \over 3})_{0}^{m}={m^{2} \over 2}+a{m^{3} \over 3}}
∫
γ
y
d
s
,
{\displaystyle \int _{\gamma }^{}yds\ ,}
γ
=
A
^
B
{\displaystyle \ \gamma ={\widehat {A}}B}
r
{\displaystyle \ r}
2
α
{\displaystyle \ 2\alpha }
y
.
{\displaystyle \ y\ .}
Si ha:
x
=
r
sin
α
,
y
=
r
cos
α
,
d
s
=
x
′
2
(
α
)
+
y
′
2
(
α
)
2
d
s
=
r
d
α
,
{\displaystyle \ x=r\sin \alpha ,\ y=r\cos \alpha ,\ ds={\sqrt[{2}]{x^{'2}(\alpha )+y^{'2}(\alpha )}}ds=rd\alpha \ ,}
Errore del parser (SVG (MathML può essere abilitato tramite plug-in del browser): risposta non valida ("Math extension cannot connect to Restbase.") dal server "http://localhost:6011/it.wikibooks.org/v1/":): {\displaystyle \int_{\gamma}^{}y ds=\int_{-\alpha}^{\alpha}r^2 \cos\alpha d\alpha=(r^2 \sin\alpha)_{-\alpha}^{\alpha}=2 r^2 \sin \alpha=r \bar {AB}\ .}
∫
∫
Ω
x
y
d
x
d
y
,
{\displaystyle \int \int _{\Omega }^{}xy\ dx\ dy\ ,}
essendo
Ω
{\displaystyle \ \Omega }
x
2
a
2
+
y
2
b
2
=
1
.
{\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}=1\ .}
Si ha quindi\ :
∬
Ω
x
y
d
x
d
y
=
∫
0
b
d
y
∫
0
a
b
b
2
−
y
2
2
x
y
d
x
=
a
2
2
b
2
∫
0
b
y
(
b
2
−
y
2
)
d
y
=
a
2
b
2
8
{\displaystyle \iint _{\Omega }^{}xy\ dx\ dy=\int _{0}^{b}dy\int _{0}^{{a \over b}{\sqrt[{2}]{b^{2}-y^{2}}}}xy\ dx={a^{2} \over 2b^{2}}\int _{0}^{b}y(b^{2}-y^{2})\ dy={a^{2}b^{2} \over 8}}
ovvero :
∬
Ω
x
y
d
x
d
y
=
∫
0
a
d
x
∫
0
b
a
a
2
−
x
2
2
x
y
d
y
=
b
2
2
a
2
∫
0
a
x
(
a
2
−
x
2
)
d
x
=
a
2
b
2
8
.
{\displaystyle \iint _{\Omega }^{}xy\ dx\ dy=\int _{0}^{a}dx\int _{0}^{{b \over a}{\sqrt[{2}]{a^{2}-x^{2}}}}xy\ dy={b^{2} \over 2a^{2}}\int _{0}^{a}x(a^{2}-x^{2})\ dx={a^{2}b^{2} \over 8\ .}}
∫
∫
∫
V
x
2
d
x
d
y
d
z
{\displaystyle \int \int \int _{V}^{}x^{2}dx\ dy\ dz}
esendo
V
{\displaystyle \ V}
A
=≡
(
a
,
o
,
o
)
,
B
≡
(
0
,
b
,
0
)
,
c
≡
(
0
,
0
,
c
)
;
{\displaystyle \ A=\equiv (a,o,o),\quad B\equiv (0,b,0),\quad c\equiv (0,0,c)\ ;}
∭
V
x
2
d
x
d
y
d
z
=
∫
0
c
d
z
∫
0
b
d
y
∫
0
a
x
2
d
x
=
1
3
∫
0
c
d
z
∫
0
b
a
3
d
y
=
a
3
b
3
∫
0
c
d
z
=
a
3
b
c
3
{\displaystyle \iiint _{V}^{}x^{2}dx\ dy\ dz=\int _{0}^{c}dz\int _{0}^{b}dy\int _{0}^{a}x^{2}dx={1 \over 3}\int _{0}^{c}dz\int _{0}^{b}a^{3}dy={a^{3}b \over 3}\int _{0}^{c}dz={a^{3}bc \over 3}}
![{\displaystyle \ x=r\sin \alpha ,\ y=r\cos \alpha ,\ ds={\sqrt[{2}]{x^{'2}(\alpha )+y^{'2}(\alpha )}}ds=rd\alpha \ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba90c756420918d1180a18275a6a7078ea9798c)
![{\displaystyle \iint _{\Omega }^{}xy\ dx\ dy=\int _{0}^{b}dy\int _{0}^{{a \over b}{\sqrt[{2}]{b^{2}-y^{2}}}}xy\ dx={a^{2} \over 2b^{2}}\int _{0}^{b}y(b^{2}-y^{2})\ dy={a^{2}b^{2} \over 8}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb251ec4b473624d12781d9acf1d5b50027727a)
![{\displaystyle \iint _{\Omega }^{}xy\ dx\ dy=\int _{0}^{a}dx\int _{0}^{{b \over a}{\sqrt[{2}]{a^{2}-x^{2}}}}xy\ dy={b^{2} \over 2a^{2}}\int _{0}^{a}x(a^{2}-x^{2})\ dx={a^{2}b^{2} \over 8\ .}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cc4d71f5f40b802305a2393d63e883f63a608da)
