Chapter 5 Integration Third big topic of calculus

Integration used to: Find area under a curve

Integration used to: Find area under a curve Find volume of surfaces of revolution

Integration used to: Find area under a curve Find volume of surfaces of revolution Find total distance traveled

Integration used to: Find area under a curve Find volume of surfaces of revolution Find total distance traveled Find total change Just to name a few

Area under a curve can be approximated without using calculus.

exact area. Then we’ll do it with calculus to find exact area.

Rectangular Approximation Method 5.1 Left Right Midpoint

5.2 Definite Integrals

Anatomy of an integral integral sign

Anatomy of an integral integral sign [a,b] interval of integration a, b limits of integration

Anatomy of an integral integral sign [a,b] interval of integration a, b limits of integration a lower limit b upper limit

Anatomy of an integral integral sign [a,b] interval of integration a, b limits of integration a lower limit b upper limit f(x) integrand x variable of integration

Rules for definite integrals If f and g are integrable functions on [a,b] and [b,c] respectively 1. Zero Rule

Rules for definite integrals If f and g are integrable functions on [a,b] and [b,c] respectively 2. Reversing limits of integration Rule

Rules for definite integrals If f and g are integrable functions on [a,b] and [b,c] respectively 3. Constant Multiple Rule

Rules for definite integrals If f and g are integrable functions on [a,b] and [b,c] respectively 4. Sum, Difference Rule

Rules for definite integrals If f and g are integrable functions on [a,b] and [b,c] respectively 6. Domination Rule 6a. Special case

Rules for definite integrals If f and g are integrable functions on [a,b] and [b,c] respectively 7. Max-Min Rule

Rules for definite integrals If f and g are integrable functions on [a,b] and [b,c] respectively 8. Interval Addition Rule

Rules for definite integrals If f and g are integrable functions on [a,b] and [b,c] respectively 9. Interval Subtraction Rule

THE FUNDAMENTAL THEOREM OF CALCULUS PART 1 THEORY PART 11 INTEGRAL EVALUATION

INTEGRAL AS AREA FINDER Area above x-axis is positive. Area below x-axis is negative. “total” area is area above – area below “net” area is area above + area below

TEST LRAM RRAM MRAM SUMMATION REIMANN SUMS RULES FOR INTEGRALS FUND. THM. CALC EVALUATE INTEGRALS FIND AREA TOTAL AREA NET AREA ETC……..