The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
①
S
=
∫
a
b
y
d
x
=
∫
a
b
f
(
x
)
d
x
{\displaystyle S=\int _{a}^{b}ydx=\int _{a}^{b}f(x)dx\,}
ជាទូទៅ
S
=
∫
a
b
|
f
(
x
)
|
d
x
{\displaystyle \color {blue}S=\int _{a}^{b}|f(x)|dx\,}
②
S
=
∫
a
b
[
f
(
x
)
−
g
(
x
)
]
d
x
{\displaystyle S=\int _{a}^{b}[f(x)-g(x)]dx\,}
ជាទូទៅ
S
=
∫
a
b
|
f
(
x
)
−
g
(
x
)
|
d
x
{\displaystyle \color {blue}S=\int _{a}^{b}|f(x)-g(x)|dx\,}
ប្រព័ន្ធសមីការ
{
x
=
f
(
t
)
y
=
g
(
t
)
{\displaystyle {\begin{cases}x=f(t)\\y=g(t)\\\end{cases}}}
S
=
∫
a
b
|
y
|
d
x
=
∫
α
β
|
y
|
d
x
d
t
d
t
{\displaystyle S=\int _{a}^{b}|y|dx\,=\int _{\alpha }^{\beta }|y|{\frac {dx}{dt}}dt}
x
a
→
b
{\displaystyle a\rightarrow b\,}
t
α
→
β
{\displaystyle \alpha \rightarrow \beta \,}
S
=
∫
α
β
1
2
r
2
d
θ
=
∫
α
β
1
2
[
f
(
θ
)
]
2
d
θ
{\displaystyle \color {blue}S=\int _{\alpha }^{\beta }{\frac {1}{2}}r^{2}d\theta \,=\int _{\alpha }^{\beta }{\frac {1}{2}}[f(\theta )]^{2}d\theta }
ក្រលាផ្ទៃនៃដែនកំនត់ D នៅលើប្លង់ xy កំនត់ដោយ
∫
D
d
S
=
∬
D
d
x
d
y
{\displaystyle \color {blue}\int _{D}dS\,=\iint _{D}dxdy\,}
V
=
∫
a
b
S
(
x
)
d
x
{\displaystyle \color {blue}V=\int _{a}^{b}S(x)dx}
មាឌនៃសូលីដវិលជុំវិញអ័ក្ស x នៅចន្លោះ
a
≤
x
≤
b
{\displaystyle a\leq x\leq b\,}
y
=
f
(
x
)
{\displaystyle y=f(x)\,}
V
=
π
∫
a
b
y
2
d
x
=
π
∫
a
b
[
f
(
x
)
]
2
d
x
{\displaystyle V=\pi \int _{a}^{b}y^{2}dx=\pi \int _{a}^{b}[f(x)]^{2}dx}
សូលីដដែលមានផ្ទៃបាតជាដែនកំនត់ D នៅលើប្លង់ xy និង ស៊ីឡាំងមានទ្រនុងស្របនឹងអ័ក្ស z
z
=
f
(
x
,
y
)
{\displaystyle z=f(x,y)\,\,}
(
f
(
x
,
y
)
≡
0
{\displaystyle (f(x,y)\equiv 0\,}
នោះគេបានមាឌនៃកំណាត់សូលីដកំនត់ដោយ
V
=
∬
D
z
d
x
d
y
=
∬
D
f
(
x
,
y
)
d
x
d
y
{\displaystyle V=\iint _{D}zdxdy=\iint _{D}f(x,y)dxdy\,}
ក្រលាផ្ទៃនៃលំហំដែលមានដែនកំនត់ D កំនត់ដោយ
V
=
∭
D
d
V
=
∭
D
d
x
d
y
d
z
{\displaystyle V=\iiint _{D}dV=\iiint _{D}dxdydz\,}
ប្រវែងអង្កត់ធ្នូនៃខ្សែកោង
y
=
f
(
x
)
{\displaystyle y=f(x)\,}
s
=
∫
a
b
1
+
(
d
y
d
x
)
2
d
x
=
∫
a
b
1
+
[
f
′
(
x
)
]
2
d
x
{\displaystyle \color {blue}s=\int _{a}^{b}{\sqrt {1+({\frac {dy}{dx}})^{2}}}\,\,\,dx=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\,\,\,dx}
អង្កត់ធ្នូនៃខ្សែកោងដែលមានរាងជាប៉ារ៉ាម៉ែត្រ ប្រព័ន្ធសមីការប៉ារ៉ាម៉ែត្រនៃខ្សែកោង
{
x
=
f
(
t
)
y
=
g
(
t
)
{\displaystyle {\begin{cases}x=f(t)\\y=g(t)\\\end{cases}}}
α
≤
t
≤
β
{\displaystyle \alpha \leq t\leq \beta \,}
គេបាន
s
=
∫
α
β
(
d
x
d
t
)
2
+
(
d
y
d
t
)
2
d
t
=
∫
α
β
[
f
′
(
t
)
]
2
+
[
g
′
(
t
)
]
2
d
t
{\displaystyle \color {blue}s=\int _{\alpha }^{\beta }{\sqrt {({\frac {dx}{dt}})^{2}+({\frac {dy}{dt}})^{2}}}\,\,\,dt=\int _{\alpha }^{\beta }{\sqrt {[f'(t)]^{2}+[g'(t)]^{2}}}\,\,\,dt}
ប្រវែងអង្កត់ធ្នូក្នុងកូអរដោនេប៉ូលែរ គេមានខ្សែកោង
r
=
f
(
θ
)
(
α
≤
θ
≤
β
)
{\displaystyle r=f(\theta )\,\,\,(\alpha \leq \theta \leq \beta )}
s
=
∫
α
β
r
2
+
(
d
r
d
θ
)
2
d
θ
{\displaystyle \color {blue}s=\int _{\alpha }^{\beta }{\sqrt {r^{2}+({\frac {dr}{d\theta }})^{2}}}\,\,\,d\theta }
![{\displaystyle S=\int _{a}^{b}[f(x)-g(x)]dx\,}](/media/api/rest_v1/media/math/render/svg/af704871ea71ee2eaad2849dfd4485655a2338f6)
![{\displaystyle \color {blue}S=\int _{\alpha }^{\beta }{\frac {1}{2}}r^{2}d\theta \,=\int _{\alpha }^{\beta }{\frac {1}{2}}[f(\theta )]^{2}d\theta }](/media/api/rest_v1/media/math/render/svg/2f26f6ba802e022b0e50b35696bbadca7eea6098)
![{\displaystyle V=\pi \int _{a}^{b}y^{2}dx=\pi \int _{a}^{b}[f(x)]^{2}dx}](/media/api/rest_v1/media/math/render/svg/0cdc5928e013755f4ffc34a22296efebb0d3116c)
![{\displaystyle \color {blue}s=\int _{a}^{b}{\sqrt {1+({\frac {dy}{dx}})^{2}}}\,\,\,dx=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\,\,\,dx}](/media/api/rest_v1/media/math/render/svg/0eeb48dc3674b02e3647c1e5b58a00f967fba089)
![{\displaystyle \color {blue}s=\int _{\alpha }^{\beta }{\sqrt {({\frac {dx}{dt}})^{2}+({\frac {dy}{dt}})^{2}}}\,\,\,dt=\int _{\alpha }^{\beta }{\sqrt {[f'(t)]^{2}+[g'(t)]^{2}}}\,\,\,dt}](/media/api/rest_v1/media/math/render/svg/f1ef0c25e110dcaf1eee69fa85c96ba51979e112)
