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The Fundamental Theorem of Calculus

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MAT137Y1 – LEC0501

Calculus!

The Fundamental Theorem of Calculus

Part 2

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For next week

For Monday (Jan 28), watch the videos:

• Integration by parts: (9.5), 9.6, (9.7), (9.8), (9.9) For Wednesday (Jan 30), watch the videos:

• Integration of trig functions: 9.10, (9.11), (9.12)

• Integration of rational functions: 9.15, (9.16), (9.17)

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What is wrong?

Let 𝑓 (𝑥) = 1

𝑥

4

and 𝐹 (𝑥) = − 1

3𝑥

3

. Notice that 𝐹 ′ = 𝑓 .

Hence, according to FTC-2, we have

∫

1

−1

1

𝑥 4 𝑑𝑥 = −1 3𝑥 3 |

1

−1 = − 2 3

However, 𝑥 4 is always positive and −1 < 1, so

the integral should be positive.

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Definite integrals

Justify that the following integrals are well defined and compute them:

1 ∫

2 1

𝑥 3 𝑑𝑥

2 ∫

1 0

[𝑒 𝑥 + 𝑒 −𝑥 − cos(2𝑥)] 𝑑𝑥

3 ∫

1/√2 1/2

4

√1 − 𝑥 2 𝑑𝑥

4 ∫

𝜋/3 𝜋/4

sec 2 𝑥 𝑑𝑥

5 ∫

2 1 [

𝑑 𝑑𝑥 (

sin 2 𝑥

1 + arctan 2 𝑥 + 𝑒 −𝑥

2

)] 𝑑𝑥

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Riemann sums

Compute the following limits

1 lim

𝑛→+∞

1 𝑛

𝑛

∑ 𝑘=1

tan 𝑘 𝑛

2 lim

𝑛→+∞

𝑛

∑ 𝑘=1

𝑛 𝑛 2 + 𝑘 2

3 lim

𝑛→+∞

𝑛

∏ 𝑘=1 (1 + 𝑘 𝑛 )

1 𝑛

Hints:

• 𝑑

𝑑𝑥 ( − ln | cos (𝑥)|) = tan(𝑥)

• 𝑑

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Area

Compute the area of the bounded region between 𝑦 = 𝑥 2

2 and 𝑦 = 1

1 + 𝑥 2 .

Références

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