THE FUNDAMENTAL THEOREM OF CALCULUS


THE FUNDAMENTAL THEOREM OF CALCULUS, PART I
If f is continuous on [a,b], then the function g defined by

g(x) = ó
õ		 x

a 
 		f(t)dt    a £ x £ b
is continuous on [a,b] and differentiable on (a,b), and g¢ (x) = f(x) .

EXAMPLES
Use Part I of the Fundamental Theorem of Calculus to find the derivative of the given function.

1.  
 
 g(x) = ó
õ		 x

1 
 		(t2 -1)20dt
2. 
 
 g(u) = ó
õ		 u

p 
 		1/(1+t4)dt
3. 
 
 y = ó
õ		 p

x2 
 		sint/t dt
4. 
 
  ó
õ		 17

tanx 
 		sin(t4)dt
THE FUNDAMENTAL THEOREM OF CALCULUS, PART II
If f is continuous on [a,b], then
ó
õ		 b

a 
 		f(x)dx = F(b)-F(a)
where F is any antiderivative of f, that is, F¢ = f .

EXAMPLES
Use Part 2 of the Fundamental Theorem of Calculus to evaluate the integral, or state that it does not exist.

1. 
 
  ó
õ		 4

-2 
 		(3x-5)dx
2. 
 
  ó
õ		 2

1 
 		x-2dx
3. 
 
  ó
õ		 2

1 
 		(5x2 -4x+3)dx
4. 
 
  ó
õ		 1

-1 
 		3/(t4)dt
For an Indefinite Integral:
ó
õ		 f(x)dx = F(x)   means   F¢(x) = f(x)
 

EXERCICES IV
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the given function.

1. 
 
 g(x) = ó
õ		 x1/2

-1 
 		(t3 +1)1/2dt
2. 
 
 f(x) = ó
õ		 2

x 
 		cos(t2)dt
3. 
 
 h(x)- ó
õ		 1/x

2 
 		sin4tdt
4. 
 
 y = ó
õ		 sinx

-5 
 		t cos(t3)dt
Use Part 2 of the Fundamental Theorem of calculus to find the derivative of the given function, or state that it does not exist.

5. 
 
  ó
õ		 0

-3 
 		(5y4 -6y2+14)dy
6. 
 
  ó
õ		 1

0 
 		(y9-2y5+3y)dy
7. 
 
  ó
õ		 p/3

p/6 
 		csc2qdq
8. 
 
  ó
õ		 p/2

p/3 
 		cscx cotx dx
9. 
 
  ó
õ		 2

1 
 		(t6 -t2)/t4 dt

 

File translated from TEX by TTH, version 2.00.
On 12 Apr 2002, 09:12.