Download Math 71 – 1.1

Math 140
5.3 – The Definite Integral and the
Fundamental Theorem of Calculus
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Q: How can we find the area under any curve?
For example, let’s try to find the area under
𝑓 𝑥 = 𝑥 3 − 9𝑥 2 + 25𝑥 − 19
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What is
the area
of this?
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1 rectangle
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2 rectangles
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4 rectangles
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50 rectangles
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If the left sides of our rectangles are at 𝑥1 , 𝑥2 , …, 𝑥𝑛
and each rectangle has width Δ𝑥,
then the areas of the rectangles are
𝑓 𝑥1 Δ𝑥, 𝑓 𝑥2 Δ𝑥, …, 𝑓 𝑥𝑛 Δ𝑥.
So, the sum of the areas of the rectangles (called a
Riemann sum) is:
𝑓 𝑥1 Δ𝑥 + 𝑓 𝑥2 Δ𝑥 + ⋯ + 𝑓 𝑥𝑛 Δ𝑥
= 𝑓 𝑥1 + 𝑓 𝑥2 + ⋯ + 𝑓 𝑥𝑛 Δ𝑥
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So, the sum of the areas of the rectangles (called a
Riemann sum) is:
𝑓 𝑥1 Δ𝑥 + 𝑓 𝑥2 Δ𝑥 + ⋯ + 𝑓 𝑥𝑛 Δ𝑥
= 𝑓 𝑥1 + 𝑓 𝑥2 + ⋯ + 𝑓 𝑥𝑛 Δ𝑥
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So, the sum of the areas of the rectangles (called a
Riemann sum) is:
𝑓 𝑥1 Δ𝑥 + 𝑓 𝑥2 Δ𝑥 + ⋯ + 𝑓 𝑥𝑛 Δ𝑥
= 𝑓 𝑥1 + 𝑓 𝑥2 + ⋯ + 𝑓 𝑥𝑛 Δ𝑥
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As the number of rectangles (𝑛) increases, the
above sum gets closer to the true area.
That is,
Area under curve =
𝒍𝒊𝒎 𝒇 𝒙𝟏 + 𝒇 𝒙𝟐 + ⋯ + 𝒇 𝒙𝒏
𝐧→+∞
𝜟𝒙
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𝑏
𝑓 𝑥 𝑑𝑥 = lim 𝑓 𝑥1 + 𝑓 𝑥2 + ⋯ + 𝑓 𝑥𝑛
𝑎
n→+∞
Δ𝑥
The above integral is called a definite integral, and 𝑎
and 𝑏 are called the lower and upper limits of
integration.
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The Fundamental Theorem of Calculus
If 𝑓(𝑥) is continuous on the interval 𝑎 ≤ 𝑥 ≤ 𝑏,
then
𝒃
𝒇 𝒙 𝒅𝒙 = 𝑭 𝒃 − 𝑭(𝒂)
𝒂
where 𝐹(𝑥) is any antiderivative of 𝑓(𝑥) on 𝑎 ≤
𝑥 ≤ 𝑏.
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Ex 1.
Evaluate the given definite integrals using the
fundamental theorem of calculus.
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2𝑥 2 𝑑𝑥
−1
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Ex 1.
Evaluate the given definite integrals using the
fundamental theorem of calculus.
1
2𝑥 2 𝑑𝑥
−1
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Ex 1 (cont).
Evaluate the given definite integrals using the
fundamental theorem of calculus.
1
(𝑒 2𝑥 − 𝑥) 𝑑𝑥
0
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Ex 1 (cont).
Evaluate the given definite integrals using the
fundamental theorem of calculus.
1
(𝑒 2𝑥 − 𝑥) 𝑑𝑥
0
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Note: Definite integrals can evaluate to
negative numbers. For example,
2
1
(1
−
𝑥)
𝑑𝑥
=
−
1
2
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Just like with indefinite integrals, terms can be integrated
separately, and constants can be pulled out.
𝑏
𝑏
𝑓 𝑥 ± 𝑔(𝑥) 𝑑𝑥 =
𝑎
𝑏
𝑏
𝑘𝑓 𝑥 𝑑𝑥 = 𝑘
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 ±
𝑎
𝑔(𝑥) 𝑑𝑥
𝑎
𝑓 𝑥 𝑑𝑥
𝑎
Also,
𝑎
𝑓 𝑥 𝑑𝑥 = 0
𝑎
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = −
𝑏
𝑓 𝑥 𝑑𝑥
𝑎
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𝑏
𝑐
𝑓 𝑥 𝑑𝑥 =
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 +
𝑎
𝑓 𝑥 𝑑𝑥
𝑐
(here, 𝑐 is any real number)
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𝑏
𝑐
𝑓 𝑥 𝑑𝑥 =
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 +
𝑎
𝑓 𝑥 𝑑𝑥
𝑐
(here, 𝑐 is any real number)
=
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𝑏
𝑐
𝑓 𝑥 𝑑𝑥 =
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 +
𝑎
𝑓 𝑥 𝑑𝑥
𝑐
(here, 𝑐 is any real number)
=
+
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Ex 2.
Let 𝑓(𝑥) and 𝑔(𝑥) be functions that are continuous on the interval
− 3 ≤ 𝑥 ≤ 5 and that satisfy
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5
𝑓 𝑥 𝑑𝑥 = 2
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𝑔 𝑥 𝑑𝑥 = 7
−3
−3
𝑓 𝑥 𝑑𝑥 = −8
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Use this information along with the rules definite integrals to evaluate
the following.
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2𝑓 𝑥 − 3𝑔 𝑥
𝑑𝑥
−3
1
𝑓 𝑥 𝑑𝑥
−3
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Ex 3.
Evaluate:
1
8𝑥 𝑥 2 + 1
3
𝑑𝑥
0
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Ex 3.
Evaluate:
1
8𝑥 𝑥 2 + 1
3
𝑑𝑥
0
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Net Change
If we have 𝑄′(𝑥), then we can calculate the net
change in 𝑄(𝑥) as 𝑥 goes from 𝑎 to 𝑏 with an
integral:
𝑏
𝑄 ′ 𝑥 𝑑𝑥
𝑄 𝑏 −𝑄 𝑎 =
𝑎
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Net Change
Ex 4.
Suppose marginal cost is 𝐶 ′ 𝑥 = 3 𝑥 − 4 2
dollars per unit when the level of production is 𝑥
units. By how much will the total manufacturing
cost increase if the level of production is raised
from 6 units to 10 units?
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