Lecture 19- First Fundamental Theorem of Calculus PDF

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Lecture 19- First Fundamental Theorem of Calculus...

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18.01 Single Variable Calculus Fall 2006

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Lecture 19

18.01 Fall 2006

Lecture 19: First Fundamental Theorem of Calculus Fundamental Theorem of Calculus (FTC 1) If f (x) is continuous and F � (x) = f (x), then � b f (x)dx = F (b) − F (a) a

�b � � �x=b Notation: F (x) � = F (x) � = F (b) − F (a) a

x=a

x3 Example 1. F (x) = , F � (x) = x2 ; 3

�

b

a

�b x3 �� b3 a3 = x dx = − 3 3 �a 3 2

Example 2. Area under one hump of sin x (See Figure 1.) � π �π � sin x dx = − cos x� = − cos π − (− cos 0) = −(−1) − (−1) = 2 0

0

1

� Figure 1: Graph of f (x) = sin x for 0 ≤ x ≤ π.

Example 3.

�

1

x5 dx = 0

�1 1 1 x 6 �� = −0 = 6 6 �0 6

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Lecture 19

18.01 Fall 2006

Intuitive Interpretation of FTC: v(t) = x� (t) =

x(t) is a position;

�

dx is the speed or rate of change of x. dt

b

v(t)dt = x(b) − x(a)

(FTC 1)

a

R.H.S. is how far x(t) went from time t = a to time t = b (difference between two odometer readings). L.H.S. represents speedometer readings. n

v (t i )∆t

�

approximates the sum of distances traveled over times ∆t

i=1

The approximation above is accurate if v (t ) is close to v (ti ) on the i th interval. The interpretation of x (t ) as an odometer reading is no longer valid if v changes sign. Imagine a round trip so that x(b) − x(a) = 0. Then the positive and negative velocities v(t) cancel each other, whereas an odometer would measure the total distance not the net distance traveled. Example 4.

�

2π 0

�2π � sin x dx = − cos x� = − cos 2π − (− cos 0) = 0. 0

The integral represents the sum of areas under the curve, above the x-axis minus the areas below the x-axis. (See Figure 2.)

1

+

2�

-

Figure 2: Graph of f (x) = sin x for 0 ≤ x ≤ 2π.

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Lecture 19

18.01 Fall 2006

Integrals have an important additive property (See Figure 3.) �

b

f (x)dx + a

�

c b

� f (x)dx =

a

c

f (x)dx a

b

c

Figure 3: Illustration of the additive property of integrals New Definition: �

a b

� f (x)dx = −

b

f (x)dx a

This definition is used so that the fundamental theorem is valid no matter if a < b or b < a. It also makes it so that the additive property works for a, b, c in any order, not just the one pictured in Figure 3.

3

Lecture 19

18.01 Fall 2006

Estimation: �

If f (x) ≤ g (x), then

b

f (x)dx ≤ a

b

�

g (x)dx (only if a < b) a

Example 5. Estimation of e x Since 1 ≤ ex for x ≥ 0, �

1

1dx ≤

0

�

1

�

1

ex dx 0

�1 � e x dx = ex � = e1 − e 0 = e − 1 0

0

Thus 1 ≤ e − 1, or e ≥ 2.

Example 6. We showed earlier that 1 + x ≤ e x. It follows that � �

1

(1 + x)dx ≤ 0 1

(1 + x)dx = 0

� �

3 5 Hence, ≤ e − 1,or, e ≥ . 2 2

1

ex dx = e − 1 0

x+

x2 2

��1 �� 3 = �0 2

Change of Variable: If f (x) = g (u(x)), then we write du = u� (x)dx and �

g (u)du =

�

g (u(x))u� (x)dx =

�

f (x)u� (x)dx

(indefinite integrals)

For definite integrals: �

Example 7.

�

2 1

x2

f (x)u� (x)dx = x1

�

u2

g(u)du where u1 = u(x1 ), u2 = u(x2 ) u1

� 3 �4 x + 2 x2 dx Let u = x3 + 2. Then du = 3x 2 dx =⇒ x2 dx =

du ; 3

x1 = 1, x2 = 2 =⇒ u1 = 13 + 2 = 3, u2 = 23 + 2 = 10, and � � 2 � 10 � 3 �4 du u5 ��10 105 − 35 x + 2 x 2 dx = u4 = = 15 3 15 �3 1 3

4...