The First Fundamental Theorem of Calculus

In this topic:

(1) Proposition (First Fundamental Theorem of Calculus) If first fundamental theorem of calculus _gr_1.gif] is a continuous function on the interval first fundamental theorem of calculus _gr_2.gif] and first fundamental theorem of calculus _gr_3.gif] is any anitderivative of first fundamental theorem of calculus _gr_4.gif] throughout the interval first fundamental theorem of calculus _gr_5.gif] then

first fundamental theorem of calculus _gr_6.gif]

(2) Example (First Fundamental Theorem of Calculus) Given first fundamental theorem of calculus _gr_7.gif], evaluate

first fundamental theorem of calculus _gr_8.gif]

(3) Example (First Fundamental Theorem of Calculus) Evaluate

first fundamental theorem of calculus _gr_9.gif]

(4) Example (First Fundamental Theorem of Calculus) Evaluate

first fundamental theorem of calculus _gr_10.gif]

(5) Example (First Fundamental Theorem of Calculus) Evaluate

first fundamental theorem of calculus _gr_11.gif]

(6) Example (First Fundamental Theorem of Calculus) Evaluate

first fundamental theorem of calculus _gr_12.gif]

(7) Example (First Fundamental Theorem of Calculus) Evaluate

first fundamental theorem of calculus _gr_13.gif]   where    first fundamental theorem of calculus _gr_14.gif]

Proposition (First Fundamental Theorem of Calculus) If first fundamental theorem of calculus _gr_15.gif] is a continuous function on the interval first fundamental theorem of calculus _gr_16.gif] and first fundamental theorem of calculus _gr_17.gif] is any anitderivative of first fundamental theorem of calculus _gr_18.gif] throughout the interval first fundamental theorem of calculus _gr_19.gif] then

first fundamental theorem of calculus _gr_20.gif]

    The hypothesis of the first fundamental theorem should not be overlooked; for example, if we apply the first fundamental theorem to evaluate

first fundamental theorem of calculus _gr_21.gif]  

then we obtain something which is obviously not true since first fundamental theorem of calculus _gr_22.gif] is never negative. Since first fundamental theorem of calculus _gr_23.gif] is not continuous on first fundamental theorem of calculus _gr_24.gif] the first fundamental theorem should not be used.

Example (First Fundamental Theorem of Calculus) Given first fundamental theorem of calculus _gr_25.gif], evaluate

first fundamental theorem of calculus _gr_26.gif]

    Solution. Since first fundamental theorem of calculus _gr_27.gif] using the fundamental theorem of calculus,  
    
first fundamental theorem of calculus _gr_28.gif]
first fundamental theorem of calculus _gr_29.gif]

Example (First Fundamental Theorem of Calculus) Evaluate

first fundamental theorem of calculus _gr_30.gif]

    Solution. Since first fundamental theorem of calculus _gr_31.gif] on first fundamental theorem of calculus _gr_32.gif] using the fundamental theorem of calculus,  
    
first fundamental theorem of calculus _gr_33.gif]
first fundamental theorem of calculus _gr_34.gif]    

Example (First Fundamental Theorem of Calculus) Evaluate

first fundamental theorem of calculus _gr_35.gif]

    Solution. Note that
    
first fundamental theorem of calculus _gr_36.gif]

using the fundamental theorem of calculus and the subdivision rule,   

first fundamental theorem of calculus _gr_37.gif]

first fundamental theorem of calculus _gr_38.gif]

first fundamental theorem of calculus _gr_39.gif]

first fundamental theorem of calculus _gr_40.gif]

first fundamental theorem of calculus _gr_41.gif]
first fundamental theorem of calculus _gr_42.gif]

Example (First Fundamental Theorem of Calculus) Evaluate

first fundamental theorem of calculus _gr_43.gif]

    Solution. Using the fundamental theorem of calculus,
    
first fundamental theorem of calculus _gr_44.gif]

first fundamental theorem of calculus _gr_45.gif]

first fundamental theorem of calculus _gr_46.gif]

first fundamental theorem of calculus _gr_47.gif]
 first fundamental theorem of calculus _gr_48.gif]

Example (First Fundamental Theorem of Calculus) Evaluate

first fundamental theorem of calculus _gr_49.gif]

    Solution. Using the fundamental theorem of calculus,

first fundamental theorem of calculus _gr_50.gif]

 first fundamental theorem of calculus _gr_51.gif]

first fundamental theorem of calculus _gr_52.gif]

first fundamental theorem of calculus _gr_53.gif]

first fundamental theorem of calculus _gr_54.gif]
 first fundamental theorem of calculus _gr_55.gif]

Example (First Fundamental Theorem of Calculus) Evaluate

first fundamental theorem of calculus _gr_56.gif]   where    first fundamental theorem of calculus _gr_57.gif]

    Solution. We break the integral up into two pieces as follows,
    
first fundamental theorem of calculus _gr_58.gif]

first fundamental theorem of calculus _gr_59.gif]

first fundamental theorem of calculus _gr_60.gif]

first fundamental theorem of calculus _gr_61.gif]

first fundamental theorem of calculus _gr_62.gif]

first fundamental theorem of calculus _gr_63.gif]
first fundamental theorem of calculus _gr_64.gif]

Cite this as:
First Fundamental Theorem Of Calculus
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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