MATH 1151 HW X31 solved worksheet PDF

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Solved sheet on Antiderivatives and its relation to the area under a graph...

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MATH 1511 : X31 Homework 𝑏 ∫𝑎 𝑓(𝑥)

Name _____________________

𝑑𝑥 represents the signed area between the curve 𝑦 = 𝑓(𝑥) and the 𝑥-axis between 𝑥 = 𝑎 and

𝑥 = 𝑏. Area above the axis is positive and area below the axis is negative. 1) To the right is the graph of 𝑦 = 𝑓(𝑥).

Shade the part of the graph measured by 2 ∫−4 𝑓(𝑥) 𝑑𝑥 . 2

Calculate ∫−4 𝑓(𝑥) 𝑑𝑥

2) To the right is the graph of 𝑦 = 𝑓(𝑥). Shade the part of the graph measured by 0 ∫−7 𝑓(𝑥) 𝑑𝑥 .

Calculate ∫−7 𝑓(𝑥) 𝑑𝑥 0

3) To the right is the graph of 𝑦 = 𝑓(𝑥).

Shade the part of the graph measured by 2 ∫−2 𝑓(𝑥) 𝑑𝑥. Calculate ∫−2 −𝑓(𝑥) 𝑑𝑥 2

4) To the right is the graph of 𝑦 = 𝑓(𝑥). Define ℎ(𝑚) by ℎ(𝑚) = 𝑓(−𝑚). Calculate ∫−4 ℎ(𝑚) 𝑑𝑚 0

5) To the right is the graph of 𝑦 = 𝑓(𝑥). Define 𝑘(𝑛) by 𝑘(𝑛) = 2𝑓(𝑛). Calculate ∫−8 𝑘(𝑛) 𝑑𝑛 −2

6) To the right is the graph of 𝑦 = 𝑓(𝑥). 𝑤

Define 𝐴(𝑤) = ∫−4 𝑓(𝑥) 𝑑𝑥 . Calculate 𝐴(−4) 𝐴(−1) 𝐴(4)

3 (−∞, −6] 𝑥 (−6, 4) 7) Define 𝐻(𝑡) by 𝐻(𝑡) = { − 2 [4, ∞) −2 0

Calculate ∫−10 𝐻(𝑡) 𝑑𝑡

1

Calculate ∫−1 𝐻(𝑡) 𝑑𝑡

Calculate ∫10 𝐻(𝑡) 𝑑𝑡 15

The left graph is 𝑦 = 𝑔(𝑥). The right graph is 𝑦 = ℎ(𝑣).

True of False. Explain your reasoning.

8) True False

9) True False

10) True False

4

4

∫ 𝑔(𝑥) 𝑑𝑥 < ∫ ℎ(𝑣) 𝑑𝑣 0

0

2

∫ 𝑔(𝑥) 𝑑𝑥 < 0 6

4

∫ ℎ(𝑣) 𝑑𝑣 < 0 0

𝑘

11) Define 𝐴(𝑘) = ∫ 𝑥 𝑑𝑥 0

Using area, create a nice formula for 𝐴(𝑘). Explain your area reasoning.

𝜋

12) Using area, explain why ∫−𝜋 sin(𝜃) 𝑑𝜃 = 0

13) Using area, explain why ∫0 cos(𝜃) 𝑑𝜃 = 0 𝜋

2

14) Using area, explain why ∫0 (𝑡 + 5)(𝑡 − 3) 𝑑𝑡 < 0

15) Suppose 𝑓(𝑥) and 𝑔(𝑥) are two functions with 𝑓(𝑥) ≥ 𝑔(𝑥) ≥ 0 for all 𝑥. 𝑏

𝑏

Using area, explain why ∫𝑎 𝑓(𝑥) 𝑑𝑥 > ∫𝑎 𝑔(𝑥) 𝑑𝑥

16) Suppose 𝑓(𝑥) and 𝑔(𝑥) are two functions with 0 ≥ 𝑓 (𝑥) ≥ 𝑔(𝑥) for all 𝑥.

Using area, explain why ∫𝑎 𝑓(𝑥) 𝑑𝑥 > ∫𝑎 𝑔(𝑥) 𝑑𝑥 𝑏

𝑏

17) Draw the graph of a function 𝐾(𝑚), such that •

•

4

∫−3 𝐾(𝑚) 𝑑𝑚 < 0 , and

∫−2 𝐾(𝑚) 𝑑𝑚 > 0 4...