The Definite Integral. In the previous section, we approximated area using rectangles with specific...
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Transcript of 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
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥
- the symbol is called an INTEGRAL sign- the numbers and are called the LIMITS of INTEGRATION
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥
- the symbol is called an INTEGRAL sign- the numbers and are called the LIMITS of INTEGRATION- the function is the INTEGRAND
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥
- the symbol is called an INTEGRAL sign- the numbers and are called the LIMITS of INTEGRATION- the function is the INTEGRAND
- ** sometimes is referred to as the LOWER LIMIT
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥
- the symbol is called an INTEGRAL sign- the numbers and are called the LIMITS of INTEGRATION- the function is the INTEGRAND
- ** sometimes is referred to as the LOWER LIMIT- ** and is referred to as the UPPER LIMIT
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥
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
The Definite Integral
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.
The Definite Integral
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
The Definite Integral
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)
We will simplify this into :
∫𝑏
𝑎
𝑓 (𝑥 )𝑑𝑥=𝑔 (𝑏)−𝑔 (𝑎)
- Where is the anti - derivatve of our function
The Definite Integral
∫1
3
(𝑥+1 )𝑑𝑥=¿¿EXAMPLE # 1 : Find
The Definite Integral
∫1
3
(𝑥+1 )𝑑𝑥=¿¿EXAMPLE # 1 : Find
Using the power rule for anti – derivatives :
The Definite Integral
∫1
3
(𝑥+1 )𝑑𝑥=¿¿EXAMPLE # 1 : Find
Using the power rule for anti – derivatives :
So now we will calculate
The Definite Integral
∫1
3
(𝑥+1 )𝑑𝑥=¿¿EXAMPLE # 1 : Find
Using the power rule for anti – derivatives :
So now we will calculate
The Definite Integral
∫1
3
(𝑥+1 )𝑑𝑥=¿¿EXAMPLE # 1 : Find
Using the power rule for anti – derivatives :
So now we will calculate
** in future examples, it is acceptable to disregard C when calculating as it will always become zero…
The Definite Integral
∫1
3
(−𝑥2+10 )𝑑𝑥=¿¿EXAMPLE # 2 : Find
The Definite Integral
∫1
3
(−𝑥2+10 )𝑑𝑥=¿¿EXAMPLE # 2 : Find
Using the power rule for anti – derivatives :
The Definite Integral
∫1
3
(−𝑥2+10 )𝑑𝑥=¿¿EXAMPLE # 2 : Find
Using the power rule for anti – derivatives :
Now calculate
The Definite Integral
∫1
3
(−𝑥2+10 )𝑑𝑥=¿¿EXAMPLE # 2 : Find
Using the power rule for anti – derivatives :
Now calculate
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥=𝑔 (𝑥)𝑏
𝑎
This is the customary notation to show
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥=𝑔 (𝑥)𝑏
𝑎
This is the customary notation to show
∫−1
3
(3 𝑥2−2𝑥+1 )𝑑𝑥EXAMPLE # 3 : Evaluate
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥=𝑔 (𝑥)𝑏
𝑎
This is the customary notation to show
∫−1
3
(3 𝑥2−2𝑥+1 )𝑑𝑥EXAMPLE # 3 : Evaluate
3
−1
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥=𝑔 (𝑥)𝑏
𝑎
This is the customary notation to show
∫−1
3
(3 𝑥2−2𝑥+1 )𝑑𝑥EXAMPLE # 3 : Evaluate
3
−1
3
−1
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥=𝑔 (𝑥)𝑏
𝑎
This is the customary notation to show
∫−1
3
(3 𝑥2−2𝑥+1 )𝑑𝑥EXAMPLE # 3 : Evaluate
3
−1
3
−1
¿ [(3)3− (3 )2+3 ]− [(−1)3− (−1 )2+(−1)]
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥=𝑔 (𝑥)𝑏
𝑎
This is the customary notation to show
∫−1
3
(3 𝑥2−2𝑥+1 )𝑑𝑥EXAMPLE # 3 : Evaluate
3
−1
3
−1
The Definite Integral
∫𝑎
𝑏
𝑓 (𝑥 )𝑑𝑥=𝑔 (𝑥)𝑏
𝑎
This is the customary notation to show
∫−1
3
(3 𝑥2−2𝑥+1 )𝑑𝑥EXAMPLE # 3 : Evaluate
3
−1
3
−1