Definite integral

1. 定積分とその基本的性質
Definite Integral and its Basic Nature
マダブ プラサド ポカレル
S.ID.:16-03-002

2. Fundamental Theorem of Calculus
微积分基本定理
• Let f be continuous real valued function(連続実
数値関数) defined on [a,b]. Define another
function F in [a,b] by
• Then F is also continuous and differentiable (連続
および微分可能)on [a,b] and
F’(x) = f(x)


x
a
dt
t
f
x
F )
(
)
(
• Every Continuous function(連続関数) f has an anitiderivative(積分)
F(x)
• Connection between derivative and integral(微分・積分)
a≤x≤b

3. Fundamental Theorem of Calculus
微积分基本定理
0
x
as
x)
(
(c)
and
0
x
as
x
c
)
(
)
(
0
lim
)
(
)
(
x
1
)
(
)
(
x
)
(
such that
x)
x
c
x
(where
c
exists
there
integral,
definite
of
M.V.T
From
)
(
1
0
lim
)
(
)
(
)
(
0
lim
)
(
)
(
)
(




















































f
f
x
f
c
f
x
x
F
dt
t
f
c
f
dt
t
f
c
f
dt
t
f
x
x
x
F
x
dt
t
f
dt
t
f
x
x
F
dt
t
f
x
F
x
x
x
x
x
x
x
x
x
x
a
x
x
a
x
a

4. 定積分(definite integral)

5. 問題
• 1.
• 2.
dx
x
x
dx
x



2
1
2
1
0
2
2
3
1
0
1
x
y Y=x2

6. 定積分の性質
(Properties of definite Integral)
定数 :constant

7. 定積分の置換積分法
(Integration by substitution of definite integral)
X a -----> b
Y α -------> β

8. 問題
• 1. Find the definite integral of

9. More examples....
1.  10
 4/3
2.