Fundamental Theorem of Calculus Basic Properties of Integrals

Fundamental Theorem of Calculus Basic Properties of Integrals Upper and Lower Estimates Intermediate Value Theorem for Integrals First Part of the Fundamental Theorem of Calculus Second Part of the Fundamental Theorem of Calculus Index FAQ

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. Index Mika Seppälä: Fundamental Theorems FAQ

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

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

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

Second Part of the Fundamental Theorem of Calculus Proof Index Mika Seppälä: Fundamental Theorems FAQ

Fundamental Theorem of Calculus We collect the previous two results into one theorem. Fundamental Theorem of Calculus Assume that f is a continuous function. Notation Index Mika Seppälä: Fundamental Theorems FAQ

Examples (1) Example Solution Index Mika Seppälä: Fundamental Theorems FAQ

Examples (2) Example Solution Index Mika Seppälä: Fundamental Theorems FAQ