Page 1 | © 2014 by Janice L. Epstein 6.2: The Fundamental Theorem of Calculus 6.2: The Fundamental Theorem of Calculus
Example: Evaluate each integral by interpreting it in terms of area.
(i)
x
x
3
ò c dt
(ii)
ò c dt
a
1
(iii)
ò t dt
0
x
Let F ( x) = ò f (u ) du , how does the area change as x changes?
a
x+h
x
ù
F ( x + h) - F ( x )
1 éê
d
= lim ê ò f (u ) du - ò f (u ) du úú
F ( x) = lim
h 0
h 0 h
dx
h
êë a
úû
a
1
= lim
h 0 h
x+h
ò
f (u ) du
x
x+h
What does
ò
f (u ) du mean graphically? It is about the area
x
under the curve f (u ) from x to x+h. That is about f ( x)⋅ h .
d
1
F ( x) = lim
h 0 h
dx
x+h
ò
x
1
f (u ) du = lim ⋅ f ( x)⋅ h = lim f ( x) = f ( x)
h 0 h
h 0
Page 2 | © 2014 by Janice L. Epstein 6.2: The Fundamental Theorem of Calculus THE FUNDAMENTAL THEOREM OF CALCULUS, PART 1
If f is continuous on [ a, b ], then the function F defined by
x
F ( x) = ò f (u ) du ,
a £ x£b
a
is continuous on [ a, b ], differentiable on (a, b) and F ¢( x) = f ( x) . So
F ( x) is an antiderivative of f ( x).
Example: Find the derivative of the given functions
x
x
2
(i) g ( x) = ò t 3 + 1 dt (ii) y = ò 2
dt
t
1
+
-1
-1
THE FUNDAMENTAL THEOREM OF CALCULUS, PART 2
b
If f is continuous on [ a, b ], then
ò
a
where F is any antiderivative of f.
f ( x) dx = F (b) - F (a ) = F ( x) a
b
Page 3 | © 2014 by Janice L. Epstein 6.2: The Fundamental Theorem of Calculus Example: Evaluate the following definite integrals
2
(i)
ò (w
3
0
-1) dw
2
p/2
0
(ii)
ò
-2
x 2 -1 dx
(iii)
ò (cos q + 2sin q ) d q
0
Page 4 | © 2014 by Janice L. Epstein 6.2: The Fundamental Theorem of Calculus If the general antiderivative of f ( x) is F ( x) + C , this can be written as
x
F ( x) + C = C + ò f (u ) du = ò f ( x ) dx
a
Where
 f  x  dx is called an indefinite integral.
Example: Compute the following indefinite integrals
(i)
(iv)
x
3/ 5
4
x
 x 5 / 3  dx
dx
 2  4x 
(ii)  cos 
 dx
5


5
(v) 
dx
2
1 x
(iii)  2e  x / 4 dx