Chapter 5

Integration

Third big topic

of calculus

Integrationused to:

Find area under a curve

Integrationused to:

Find area under a curve

Find volume of surfaces of revolution

Integrationused to:

Find area under a curve

Find volume of surfaces of revolution

Find total distance traveled

Integrationused 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 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 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 FUNDAMENTALTHE FUNDAMENTALTHEOREM OF CALCULUSTHEOREM 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 5.1-5.4

LRAM

RRAM

MRAM

SUMMATION

REIMANN SUMS

RULES FOR INTEGRALS

FUND. THM. CALC

EVALUATE INTEGRALS

FIND AREA

TOTAL AREA

NET AREA

ETC……..