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This chapter delves into the essential concept of integration in calculus, exploring its applications such as finding areas under curves, volumes of surfaces of revolution, total distance traveled, and overall changes. It illustrates the importance of definite integrals and provides insight into their anatomy, including integral signs and limits of integration. Key rules for definite integrals, such as the Zero Rule and Constant Multiple Rule, as well as the Fundamental Theorem of Calculus, are also explained, equipping learners with foundational knowledge for evaluating integrals effectively.
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Chapter 5Integration 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 approximatedwithout using calculus.
Rectangular Approximation Method5.1 • Left • Right • Midpoint
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 FUNDAMENTALTHEOREM 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
LRAM RRAM MRAM SUMMATION REIMANN SUMS RULES FOR INTEGRALS FUND. THM. CALC EVALUATE INTEGRALS FIND AREA TOTAL AREA NET AREA ETC…….. TEST 5.1-5.4
