benginning calculus lecture notes 13 - fundamental theorem of calculus 1 & 2
TRANSCRIPT
Beginning Calculus- The Fundamental Theorem of Calculus -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 1 / 15
FTC1 FTC2
Learning Outcomes
State and apply FTC1 and FTC2
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 2 / 15
FTC1 FTC2
The First Fundamental Theorem of Calculus (FTC1)
Theorem 1 (FTC1)
Let f be a continuous and integrable function on [a, b] . For x ∈ [a, b] ,define a function
F (x) =∫ xaf (t) dt
Then F is continuous on [a, b] and differentiable on (a, b) , and
F ′ (x) = f (x)
That is, F is an antiderivative of f .
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 3 / 15
FTC1 FTC2
The First Fundamental Theorem of Calculus (FTC1)
The function F depends onlyon x .
The variable of integration,t, is called a dummy variable.
Using Leibniz notation forderivative, we write
F ′ (x) =ddx
∫ xaf (t) dt = f (x)
ba
y
t
()tfy =
area = F(x)
x
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 4 / 15
FTC1 FTC2
Example
Let F (x) =∫ x0f (t) dt where
the function f is given on theright. Then,
F (0) = 0,F (1) =2,F (2) = 5,F (3) =7,F (6) = 3.
F is increasing on (0, 3) .
F has a maximum value atx = 3.
5
1
x
y
f
0
4
2 3 4 6 7
32
1
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 5 / 15
FTC1 FTC2
Example - continue
Rough sketch of the graph of F
5
1
x
y
f
0
4
2 3 4 6 7
32
1
F
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 6 / 15
FTC1 FTC2
The Second Fundamental Theorem of Calculus (FTC2)
Theorem 2 (FTC2)
If F ′ (x) = f (x) , then∫ baF ′ (x) dx =
∫ baf (x) dx
= F (b)− F (a) = F (x)|ba
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 7 / 15
FTC1 FTC2
Example
If F (x) =xn+1
n+ 1, then F ′ (x) = xn and so
∫ baF ′ (x) dx =
∫ baxndx =
bn+1
n+ 1− an+1
n+ 1=
(bn+1
)−(an+1
)n+ 1
If n = 2, then ∫ bax2dx =
b3 − a33
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 8 / 15
FTC1 FTC2
Example
Area under one hump of sin x .
x
y
∫ π
0sin xdx = (− cos x)|π0 = − cosπ + cos 0 = 2
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 9 / 15
FTC1 FTC2
Intuitive Interpretation of FTC2
x (t) is the position at time t.
x ′ (t) =dxdt= v (t) is the speed.
∫ ba
v (t)︸︷︷︸speedometer
dt = x (b)− x (a)︸ ︷︷ ︸distance travelled
(odometer)
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 10 / 15
FTC1 FTC2
Change of Variables - Substitution
Theorem 3 (Change of Variables)
Let u = u (x). Then,du = u′ (x) dx∫ x2
x1f [u (x)] u′ (x) dx =
∫ u(x2)u(x1)
f (u) du
Only works when u′ (x) does not change sign. (i.e the functionincrease or decrease steadily).
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 11 / 15
FTC1 FTC2
Example
∫ 21
(x3 + 2
)5x2dx
Let u = x3 + 2, then du = 3x2dx .
x1 = 1, u (1) = 3; x2 = 2, u (2) = 10∫ 21
(x3 + 2
)5x2dx =
13
∫ 103u5du
=118
(u6)∣∣∣103
=118
(106 − 36
)
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 12 / 15
FTC1 FTC2
Example - WARNING
∫ 1−1x2dx
Let u = x2, then du = 2xdx
x1 = −1, u (−1) = 1; x2 = 1, u (1) = 1∫ 1−1x2dx =
∫ 11
u2√udu = 0, NOT TRUE
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 13 / 15
FTC1 FTC2
Integration by Parts
Theorem 4 (Integration By Parts)
If u and v are continuous functions on [a, b] and differentiable on (a, b) ,and if u′ and v ′ are integrable on [a, b] , then∫ b
au (x) v ′ (x) dx +
∫ bau′ (x) v (x) dx = u (b) v (b)− u (a) v (a) (1)
In Liebniz notation, we normally simplify (1) as∫udv +
∫vdu = uv
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 14 / 15
FTC1 FTC2
Integration by Parts
Proof:Let F = uv , then F ′ = uv ′ + u′v . It can be shown that F ′ is integrable.Then by FTC2,∫ b
aF ′ (x) dx = F (b)− F (a) = u (b) v (b)− u (a) v (a)
VillaRINO DoMath, FSMT-UPSI
(I2) FTC1 and FTC2 15 / 15