Fundamental Theorems of Calculus

1. Fundamental Theorems of Calculus Basic Properties of Integrals Upper and Lower Estimates Intermediate Value Theorem for Integrals First Fundamental Theorem of Calculus Second Fundamental Theorem of Calculus

2. Basic Properties of Integrals Through this section we assume that all functions are continuous on a closed interval I = [a,b]. Below r is a real number, f and g are functions. Basic Properties of Integrals 1 2 4 3 5 These properties of integrals follow from the definition of integrals as limits of Riemann sums. Mika Seppälä: Fundamental Theorems

3. The rectangle bounded from above by the red line is contained in the domain bounded by the graph of g. The rectangle bounded from above by the green line contains the domain bounded by the graph of g. Upper and Lower Estimates Theorem 1 Especially: Mika Seppälä: Fundamental Theorems

4. Intermediate Value Theorem for Integrals Theorem 2 Proof By the previous theorem, By the Intermediate Value Theorem for Continuous Functions, This proves the theorem. Mika Seppälä: Fundamental Theorems

5. By the properties of integrals. By the Intermediate Value Theorem for Integrals First Fundamental Theorem of Calculus First Fundamental Theorem of Calculus Proof Mika Seppälä: Fundamental Theorems

6. Second Fundamental Theorem of Calculus Second Fundamental Theorem of Calculus Proof Mika Seppälä: Fundamental Theorems