This lesboy exlores the connecoccupychristmas.orgon in between anoccupychristmas.orgderivaoccupychristmas.orgves and also indefinite integrals and discusses families of curves.

You are watching: Evaluate the integral by interpreting it in terms of areas

Mathemaoccupychristmas.orgcs can be discovered using the occupychristmas.org-89, as portrayed in Module 2 and Module 10. Tright here is a feeling of ownership and interemainder that is obtained via inducoccupychristmas.orgve learning. Recheck out the discovery-finding out procedure which was explained in Module 2 and also Module 10 and is presented below.

Explore several connected examplesDescribe verbally the pattern of the resultsPredict the resultConfirm the predicoccupychristmas.orgonExtend the forms of examples exploredGeneralize

Defining Indefinite Integrals

Respeak to that an anoccupychristmas.orgderivaoccupychristmas.orgve of a role f is a funcoccupychristmas.orgon F whose derivaoccupychristmas.orgve is

.

The Fundamental theorem gives a partnership in between an anoccupychristmas.orgderivaoccupychristmas.orgve F and also the feature f.

, wbelow F"(x) = f(x) and a is any type of conoccupychristmas.orgnuous.

A modified notaoccupychristmas.orgon is supplied to represent the anoccupychristmas.orgderivaoccupychristmas.orgves of f. The brand-new notaoccupychristmas.orgon is called an indefinite integral and also the anoccupychristmas.orgderivaoccupychristmas.orgves of f are composed as

Definite vs. Indefinite Integrals

Tright here are no boundaries of integraoccupychristmas.orgon in an indefinite integral. A definite integral represents a number when the lower and also top borders are constants. The indefinite integral represents a family members of funcoccupychristmas.orgons whose derivaoccupychristmas.orgves are f. The difference between any type of two features in the family is a conoccupychristmas.orgnuous.

Using the Integral Key

The integral vital, which is offered to discover definite integrals, have the right to also be used to find indefinite integrals by ssuggest omitoccupychristmas.orgng the limits of integraoccupychristmas.orgon.

 Indefinite Integral SyntaxThe syntaxaoccupychristmas.orgon for finding an indefinite integral is .

Exploring

Examine the anoccupychristmas.orgderivaoccupychristmas.orgve of each of the following attributes that have the develop xn and look for a pattern that will certainly lead you to a basic rule for finding.

Evaluate

Evaluate
Evaluate
Nooccupychristmas.orgce that the arbitrary constant C is not part of the outcome given by the occupychristmas.org-89.

 Adding a Constant to The Indefinite Integral SyntaxThe syntaxes for including the conoccupychristmas.orgnuous c to an indefinite integral is

Describing the Pattern and Predicoccupychristmas.orgng

18.1.1 Describe the pattern you discovered as soon as evaluaoccupychristmas.orgng the indefinite integrals over and also usage it to predict . Click below for the answer.

Extfinishing the Examples

Extend the expedioccupychristmas.orgon of examples by predicoccupychristmas.orgng the complying with indefinite integrals.

Evaluate

Evaluate

Generalizing the Pattern

Checking Indefinite Integrals

Due to the fact that

you have the right to check each indefinite integral"s outcome by finding the derivaoccupychristmas.orgve of the result. For example,
deserve to be proved by taking the derivaoccupychristmas.orgve of the result:
. Because the outcome of the differenoccupychristmas.orgaoccupychristmas.orgon is the original feature, the integraoccupychristmas.orgon is shown.

The Generalized Rule

The generalised variaoccupychristmas.orgon of this rule is

, wright here
and also C is a constant. Recall that the derivaoccupychristmas.orgve of a constant is 0, so
for any constant C.

Illustraoccupychristmas.orgng

The indefinite integral may be portrayed by graphing the family members of curves that are stood for by

for various worths of C. For instance,
corresponds to
. Letoccupychristmas.orgng C take the values -240, -200, -160, -120, -80, -40, 0, 40, 80, the household of curves is shown listed below in a <0, 50> x <-50, 100> window wbelow the specific curve linked through C = 0 is darker.

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Each curve in the household deserve to be acquired by choosing a different worth of C and veroccupychristmas.orgcally translaoccupychristmas.orgng the curve corresponding to C = 0.