MAT 1221 Survey of Calculus Section 6.4 Area and the Fundamental Theorem of Calculus

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Presentation transcript:

MAT 1221 Survey of Calculus Section 6.4 Area and the Fundamental Theorem of Calculus

Quiz 8 minutes

Major Themes in Calculus

We do not like to use the definition Develop techniques to deal with different functions

Major Themes in Calculus

We do not like to use the definition Develop techniques to deal with different functions

Preview

Key Pay attention to the overall ideas Pay less attention to the details – We are going to use a formula to compute the definite integrals, not limits.

Example 0

Use left hand end points to get an estimation

Example 0 Use right hand end points to get an estimation

Example 0 Observation: What happen to the estimation if we increase the number of subintervals?

In General i th subinterval sample point

In General

sample point

In General Sum of the area of the rectangles is Riemann Sum

In General Sum of the area of the rectangles is Sigma Notation for summation

In General Sum of the area of the rectangles is Index Initial value (lower limit) Final value (upper limit)

In General Sum of the area of the rectangles is

Definition

upper limit lower limit integrand

Definition Integration : Process of computing integrals

Remarks We are not going to use this limit definition to compute definite integrals. We are going to use antiderivative (indefinite integral) to compute definite integrals.

Area and Indefinite Integrals

Otherwise, the definite integral may not have obvious geometric meaning.

Example 1 Compute by interpreting it in terms of area.

Example 1 We are going to use this example to verify our next formula.

Fundamental Theorem of Calculus

Remarks To simplify the computations, we always use the antiderivative with C=0.

Remarks To simplify the computations, we always use the antiderivative with C=0. We will use the following notation to stand for F(b)-F(a):

FTC

Example 2

Example 3

Example 4

The Substitution Rule for Definite Integrals For complicated integrands, we use a version of the substitution rule.

The Substitution Rule for Definite Integrals The procedures for indefinite and definite integrals are similar but different. We need to change the upper and lower limits when using a substitution. Do not change back to the original variable.

The Substitution Rule for Definite Integrals

Example 5

Example 6

Physical Meanings of Definite Integrals We will not have time to discuss the exact physical meanings. Basic Idea: The definite integral of rate of change is the net change.

Example 7 (HW 18)