Rules for Integration, Antidifferentiation Section 5.3a.

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Presentation transcript:

Rules for Integration, Antidifferentiation Section 5.3a

Consider the “Do Now”… What happens to an integral value if we simply switch the order of the limits of integration??? If we sum rectangles moving from b to a, rather than moving from a to b… The widths of the rectangles will be negative!!! This idea leads to our first new rule for definite integrals…others are intuitively true (the proofs are beyond the scope of this course…)

Rules for Definite Integrals 1. Order of Integration 2. Zero 3. Constant Multiple 4. Sum and Difference

Rules for Definite Integrals 5. Additivity 6. Max-Min Inequality: If max f and min f are the maximum and minimum values of f on [a, b], then 7. Domination

Guided Practice Suppose Find each of the following integrals, if possible. Not enough informationgiven!!!

Guided Practice Suppose Find each of the following integrals, if possible. Not enough information given!!! Not enough information given!!!

Guided Practice Suppose that Find each of the following integrals, if possible.

Connecting Differential and Integral Calculus As we have previously discussed: This means that the integral is an antiderivative of f… If F is any antiderivative of f, then for some constant C Setting x in this equation equal to a: Substitute!!!

Connecting Differential and Integral Calculus We can now evaluate the definite integral of f from a to any number x simply by computing F(x) – F(a), where F is any antiderivative of f !!!

Guided Practice Evaluate the integral: First, we need an antiderivative of the integrand: Now, use our new formula:

Guided Practice Evaluate the given integral. An antiderivative for the integrand:

Guided Practice Evaluate the given integral. An antiderivative for the integrand:

Guided Practice Evaluate the given integral. An antiderivative for the integrand:

Guided Practice Find the total shaded area Antiderivative of the function: Area of

Guided Practice Find the total shaded area Antiderivative of the function: Area of Total Shaded Area: