The Definite Integral. In the previous section, we approximated area using rectangles with specific...

The Definite Integral

∫𝑎

𝑏

𝑓 (𝑥 )𝑑𝑥

- the symbol is called an INTEGRAL sign


∫𝑎

𝑏


- the symbol is called an INTEGRAL sign- the numbers and are called the LIMITS of INTEGRATION


∫𝑎

𝑏


- the symbol is called an INTEGRAL sign- the numbers and are called the LIMITS of INTEGRATION- the function is the INTEGRAND


∫𝑎

𝑏



- ** sometimes is referred to as the LOWER LIMIT


∫𝑎

𝑏



- ** sometimes is referred to as the LOWER LIMIT- ** and is referred to as the UPPER LIMIT


∫𝑎

𝑏


Definition : Let be continuous and nonnegative on , and let be the region between the graph of and the axis on . The AREA of is defined by :

0𝑥

𝑦

𝑅

𝑎 𝑏

𝑓


In the previous section, we approximated area using rectangles with specific widths. If we could fit thousands of “partitions” ( rectangles with equal width ) whose width would approach zero into our curve, we would get a very good approximation of the area under this curve.



Hence, we could use a summation notation to show this :

∫𝑏

𝑎

𝑓 (𝑥 )𝑑𝑥=∑𝑖=1

𝑛

𝑓 (𝑥 𝑖 ) (△ 𝑥 𝑖)

- as the largest subinterval approaches a zero width



Hence, we could use a summation notation to show this :

∫𝑏

𝑎

𝑓 (𝑥 )𝑑𝑥=∑𝑖=1

𝑛

𝑓 (𝑥 𝑖 ) (△ 𝑥 𝑖)

We will simplify this into :

∫𝑏

𝑎

𝑓 (𝑥 )𝑑𝑥=𝑔 (𝑏)−𝑔 (𝑎)

- Where is the anti - derivatve of our function


∫1

3

(𝑥+1 )𝑑𝑥=¿¿EXAMPLE # 1 : Find


∫1

3


Using the power rule for anti – derivatives :


∫1

3



So now we will calculate


∫1

3



So now we will calculate

** in future examples, it is acceptable to disregard C when calculating as it will always become zero…


∫1

3

(−𝑥2+10 )𝑑𝑥=¿¿EXAMPLE # 2 : Find


∫1

3




∫1

3



Now calculate


∫𝑎

𝑏

𝑓 (𝑥 )𝑑𝑥=𝑔 (𝑥)𝑏

𝑎

This is the customary notation to show


∫𝑎

𝑏


𝑎


∫−1

3

(3 𝑥2−2𝑥+1 )𝑑𝑥EXAMPLE # 3 : Evaluate


∫𝑎

𝑏


𝑎


∫−1

3


3

−1


∫𝑎

𝑏


𝑎


∫−1

3


3

−1

3

−1


∫𝑎

𝑏


𝑎


∫−1

3


3

−1

3

−1

¿ [(3)3− (3 )2+3 ]− [(−1)3− (−1 )2+(−1)]


∫𝑎

𝑏


𝑎


∫−1

3


3

−1

3

−1

The Definite Integral. In the previous section, we approximated area using rectangles with specific...

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