Properties of Integrals. Mika Seppälä: Properties of Integrals Basic Properties of Integrals Through this section we assume that all functions are continuous.

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Presentation transcript:

Properties of Integrals

Mika Seppälä: Properties of Integrals 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 These properties of integrals follow from the definition of integrals as limits of Riemann sums.

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

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

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

Mika Seppälä: Properties of Integrals Second Fundamental Theorem of Calculus Theorem 6