Chapter 5

Integration

Third big topicof calculus

Integrationused to:

Find area under a curve

Integrationused to:

Find area under a curveFind volume of surfaces of revolution

Integrationused to:

Find area under a curveFind volume of surfaces of revolutionFind total distance traveled

Integrationused to:

Find area under a curveFind volume of surfaces of revolutionFind total distance traveledFind total change

Just to name a few

Area under a curvecan be approximated

without using calculus.

Then we’ll do itwith calculus

to find exact area.exact area.

Rectangular Approximation Method5.1

Left

Right

Midpoint

5.2 Definite Integrals

Anatomy of an integral integral sign

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

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

Anatomy of an integral integral sign[a,b] interval of integrationa, b limits of integrationa lower limitb upper limitf(x) integrandx variable of integration

Rules for definite integralsIf f and g are integrable functions on [a,b] and [b,c] respectively

1. Zero Rule

Rules for definite integralsIf f and g are integrable functions on [a,b] and [b,c] respectively

2. Reversing limits of integration Rule

Rules for definite integralsIf f and g are integrable functions on [a,b] and [b,c] respectively

3. Constant Multiple Rule

Rules for definite integralsIf f and g are integrable functions on [a,b] and [b,c] respectively

4. Sum, Difference Rule

Rules for definite integralsIf f and g are integrable functions on [a,b] and [b,c] respectively

6. Domination Rule

6a. Special case

Rules for definite integralsIf f and g are integrable functions on [a,b] and [b,c] respectively

7. Max-Min Rule

Rules for definite integralsIf f and g are integrable functions on [a,b] and [b,c] respectively

8. Interval Addition Rule

Rules for definite integralsIf f and g are integrable functions on [a,b] and [b,c] respectively

9. Interval Subtraction Rule

THE FUNDAMENTALTHE FUNDAMENTALTHEOREM OF CALCULUSTHEOREM OF CALCULUS

PART 1 THEORY

PART 11 INTEGRAL EVALUATION

INTEGRAL AS AREA FINDERArea 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 5.1-5.4LRAMRRAMMRAMSUMMATIONREIMANN SUMSRULES FOR INTEGRALS

FUND. THM. CALCEVALUATE INTEGRALSFIND AREATOTAL AREANET AREAETC……..