MATH1013 Calculus I

Tutorial 11 Anti-derivatives, Reimann Sum and Fundamental Theorem of Calculus

1) Antiderivatives

2) Table of Formulas for Indefinite integral

3) Reimann Sum

4) Special Sums

Let 𝒏 be a positive number

∑ 𝒄 = 𝒄 𝒏𝒏𝒌=𝟏 ∑ 𝒌 =

𝒏 (𝒏+𝟏)

𝟐

𝒏𝒌=𝟏

∑ 𝒌𝟐 =𝒏 (𝒏+𝟏)(𝟐𝒏+𝟏)

𝟔

𝒏𝒌=𝟏 ∑ 𝒌𝟑 =

𝒏𝟐 (𝒏+𝟏)𝟐

𝟒

𝒏𝒌=𝟏

5) Definite Integral

6) Properties of Integrals

(a) ∫ 𝒇(𝒙)𝒅𝒙 = − ∫ 𝒇(𝒙)𝒅𝒙𝒂

𝒃

𝒃

𝒂

(b) ∫ 𝒇(𝒙)𝒅𝒙 = 𝟎𝒂

𝒂

(c) ∫ 𝒇(𝒙)𝒅𝒙 = ∫ 𝒇(𝒙)𝒅𝒙𝒄

𝒂

𝒃

𝒂+ ∫ 𝒇(𝒙)𝒅𝒙

𝒃

𝒄

7) Fundamental Theorem of Calculus

Example 1 : Evaluate ∫ √𝟐𝒙 + 𝟏𝟒

𝟎𝒅𝒙 .

Example 2: Calculate ∫ 𝒍𝒏𝒙

𝒙

𝒆

𝟏𝒅𝒙 .

8) Integration of Symmetric Functions

Exercises :

1) Evaluate the following integrals :

(a) ∫(𝟑𝒙𝟓 − 𝟓𝒙𝟗)𝒅𝒙 (b) ∫ (𝟒√𝒙 −𝟒

√𝒙) 𝒅𝒙

(c) ∫ (𝟏

𝒙 √𝒙𝟐−𝟐𝟓) 𝒅𝒙 (d) ∫ (

𝟏𝟐𝒕𝟖−𝒕

𝒕𝟑 ) 𝒅𝒕

(e) ∫(𝒔𝒊𝒏𝟐𝒚 + 𝒄𝒐𝒔𝟑𝒚)𝒅𝒚 (f) ∫(𝒔𝒆𝒄𝟐𝜽 + 𝒔𝒆𝒄𝜽 𝒕𝒂𝒏𝜽)𝒅𝜽

(g) ∫(𝒔𝒆𝒄 𝟒𝜽 𝒕𝒂𝒏 𝟒𝜽)𝒅𝜽 (h) ∫ 𝒆𝒙+𝟐 𝒅𝒙

(i) ∫ √𝒙 (𝟐𝒙𝟔 − 𝟒√𝒙𝟑

)𝒅𝒙 (j) ∫𝟐+𝒙𝟐

𝟏+𝒙𝟐 𝒅𝒙

2) Given the following velocity functions of an object moving along a line, find the

position function with the given initial position. Then graph both the velocity and

position functions.

(a) 𝒗(𝒕) = 𝟔𝒕𝟐 + 𝟒𝒕 − 𝟏𝟎 ; 𝒔(𝟎) = 𝟎

(b) 𝒗(𝒕) = 𝟐 𝒄𝒐𝒔 𝒕 ; 𝒔(𝟎) = 𝟎

3) Evaluate 𝐥𝐢𝐦𝒏→∞ {𝟏

𝒏+𝟏 +

𝟏

𝒏+𝟐 + ⋯

𝟏

𝒏+𝒏 } .

4) Prove that 𝐥𝐢𝐦𝒏→∞𝟏

𝒏 {𝒔𝒊𝒏

𝒕

𝒏+ 𝒔𝒊𝒏

𝟐𝒕

𝒏+ ⋯ + 𝒔𝒊𝒏

(𝒏−𝟏) 𝒕

𝒏} =

𝟏− 𝒄𝒐𝒔 𝒕

𝒕

5) Evaluate the following limit by identifying the integral that it represents :

𝐥𝐢𝐦𝒏→∞

∑ [(𝟒𝒌

𝒏)

𝟖

+ 𝟏]

𝒏

𝒌=𝟏

(𝟒

𝒏)

6) Fundamental Theorem of Calculus:

(a) 𝒅

𝒅𝒙 ∫ 𝒔𝒊𝒏𝟐 𝒕

𝒙

𝟏𝒅𝒕 (b)

𝒅

𝒅𝒙 ∫ √𝒕𝟐 + 𝟏

𝟓

𝒙𝒅𝒕 (c)

𝒅

𝒅𝒙 ∫ 𝒄𝒐𝒔 𝒕𝟐𝒙𝟐

𝟎𝒅𝒕

(d) 𝒅

𝒅𝒙 ∫ √𝟏 + 𝒕𝟐𝒙

−𝒙𝒅𝒕 (e)

𝒅

𝒅𝒙 ∫ 𝒍𝒏 𝒕𝟐𝒆𝟐𝒙

𝒆𝒙 𝒅𝒕

(f) 𝒅

𝒅𝒙 ∫

𝒅𝒛

𝒛𝟐+𝟏

𝟏𝟎

𝒙𝟐 (g) 𝒅

𝒅𝒙 ∫ (𝒕𝟒 + 𝟔)

𝒄𝒐𝒔 𝒙

𝟏𝒅𝒕

7) Even and Odd Functions

(a) ∫ (𝒕𝒂𝒏 𝒙) 𝒅𝒙𝝅

𝟒−𝝅

𝟒

(b) ∫𝒙𝟑−𝟒𝒙

𝒙𝟐+𝟏 𝒅𝒙

𝟐

−𝟐

8) Integration by Substitutions

(a) ∫ 𝒙𝟑 (𝒙𝟒 + 𝟏𝟔)𝟔 𝒅𝒙 (b) ∫𝟐

𝒙 √𝟒𝒙𝟐−𝟏 𝒅𝒙 , 𝒙 > 𝟐 .

(c) ∫(𝒙 + 𝟏) √𝟑𝒙 + 𝟐 𝒅𝒙 (d) ∫ (𝒔𝒊𝒏 𝒙

𝒄𝒐𝒔𝟐 𝒙) 𝒅𝒙

𝝅

𝟒𝟎

(e) ∫ 𝒙𝟐 𝒆𝒙𝟑+𝟏 𝒅𝒙𝟐

−𝟏 (f) ∫ (

√𝒙+𝟏

𝟐√𝒙)

𝟒

𝒅𝒙

(g) ∫ (𝒔𝒊𝒏 𝟐𝒚

𝒔𝒊𝒏𝟐𝒚+𝟐) 𝒅𝒚

𝝅

𝟔𝟎

(Hint: 𝒔𝒊𝒏𝟐𝒚 = 𝟐 𝒔𝒊𝒏𝒚 𝒄𝒐𝒔𝒚)

(h) ∫𝒍𝒏 𝒙

𝒙

𝒆𝟐

𝟏 𝒅𝒙

(i) ∫𝒅𝒙

√𝟏+√𝟏+𝒙 (Hint: Begin with 𝒖 = √𝟏 + 𝒙 )

9) (Briggs Revision Ex.5.3 #11) Area functions

The graph of 𝒇 is shown in the figure. Let 𝑨(𝒙) = ∫ 𝒇(𝒕)𝒅𝒕𝒙

−𝟐 and 𝑭(𝒙) = ∫ 𝒇(𝒕)𝒅𝒕

𝒙

𝟒

be two area functions for 𝒇. Evaluate the following area functions.

(a) 𝑨(−𝟐) (b) 𝑭(𝟖) (c) 𝑨(𝟒) (d) 𝑭(𝟒) (e) 𝑨(𝟖)

10) (Briggs Revision Ex.5.3 #12) Area functions The

graph of 𝒇 is shown in the figure. Let be two area functions 𝑨(𝒙) =

∫ 𝒇(𝒕)𝒅𝒕𝒙

𝟎 and 𝑭(𝒙) = ∫ 𝒇(𝒕)𝒅𝒕

𝒙

𝟐 for 𝒇. Evaluate the following

area functions.

(a) 𝑨(𝟐) (b) 𝑭(𝟓) (c) 𝑨(𝟎) (d) 𝑭(𝟖) (e) 𝑨(𝟖)

(f) 𝑨(𝟓) (g) 𝑭(𝟐)

Q11-12 Definite integrals (Briggs Ex.5.3 #23-24) Evaluate the

following integrals using the Fundamental Theorem of Calculus. Discuss whether your result is

consistent with the figure.

11) ∫ (𝒙𝟐 − 𝟐𝒙 + 𝟑) 𝒅𝒙𝟏

𝟎 12) ∫ (𝒔𝒊𝒏 𝒙 + 𝒄𝒐𝒔 𝒙) 𝒅𝒙

𝟕𝝅

𝟒−𝝅

𝟒

For Q13-18 (Briggs Revision Ex.5 #8-11) Limit definition of the definite

integral Use the limit definition of the definite integral with right Riemann sums and

a regular partition

∫ 𝐟(𝐱) 𝐝𝐱𝐛

𝐚= 𝐥𝐢𝐦

𝐧→∞ ∑ [𝐟(𝐱𝐤

∗)]𝐧𝐤=𝟏 (∆𝐱)

to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer.

13) ∫ (𝟒𝒙 − 𝟐) 𝒅𝒙𝟏

𝟎 14) ∫ (𝒙𝟐 − 𝟒) 𝒅𝒙

𝟐

𝟎

15) ∫ (𝟑𝒙𝟐 + 𝒙) 𝒅𝒙𝟐

𝟏 16) ∫ (𝒙𝟑 − 𝒙) 𝒅𝒙

𝟒

𝟎

(Briggs Revision Ex.5 #15-30)

Evaluating integrals Evaluate the following integrals.

17) ∫ (𝟑𝒙𝟒 − 𝟐𝒙 + 𝟏) 𝒅𝒙𝟐

−𝟐 18) ∫ 𝒄𝒐𝒔 𝟑𝒙 𝒅𝒙

19) ∫ (𝒙 + 𝟏) 𝟑𝒅𝒙𝟐

𝟎 20) ∫ (𝟒𝒙𝟐𝟏 − 𝟐𝒙𝟏𝟔 + 𝟏) 𝒅𝒙

𝟏

𝟎

21) ∫ (𝟗𝒙𝟖 − 𝟕𝒙𝟔) 𝒅𝒙𝟏

𝟎 22) ∫ (𝒆𝟒𝒙+𝟖) 𝒅𝒙

𝟐

−𝟐

23) ∫ √𝒙 (√𝒙 + 𝟏) 𝒅𝒙𝟏

𝟎 24) ∫

𝒙𝟐

𝒙𝟑+𝟐𝟕 𝒅𝒙

25) ∫ 𝒅𝒙

√𝟒−𝒙𝟐

𝟏

𝟎 26) ∫ 𝒚𝟐 (𝟑𝒚𝟑 + 𝟏)𝟒 𝒅𝒚

27) ∫𝒙

√𝟐𝟓−𝒙𝟐 𝒅𝒙

𝟑

𝟎 28) ∫ (𝟏 − 𝒄𝒐𝒔𝟐 𝟑𝜽) 𝒅𝜽

𝝅

𝟎

29) ∫ 𝒙 𝒔𝒊𝒏 𝒙𝟐 𝒄𝒐𝒔𝟖 𝒙 𝒅𝒙 30) ∫ 𝒔𝒊𝒏𝟐 𝟓𝜽 𝒅𝜽

31) ∫𝒙𝟐+𝟐𝒙−𝟐

𝒙𝟑+𝟑𝒙𝟐−𝟔𝒙 𝒅𝒙 32) ∫ (

𝒆𝒙

𝟏+𝒆𝟐𝒙)𝒍𝒏 𝟐

𝟎 𝒅𝒙

33) (Briggs Revision Ex.5 #38) Properties of integrals

The figure shows the areas of regions bounded by the graph of 𝒇 and the 𝒙-axis. Evaluate

the following integrals.

(a) ∫ 𝒇(𝒙) 𝒅𝒙𝒄

𝒂 (b) ∫ 𝒇(𝒙) 𝒅𝒙

𝒅

𝒃 (c) ∫ 𝟐 𝒇(𝒙) 𝒅𝒙

𝒃

𝒄

(d) ∫ 𝟒 𝒇(𝒙) 𝒅𝒙𝒅

𝒂 (e) ∫ 𝟑 𝒇(𝒙) 𝒅𝒙

𝒃

𝒂 (f) ∫ 𝟐 𝒇(𝒙) 𝒅𝒙

𝒅

𝒃

34) (Briggs Revision Ex.5 #52) Identifying functions Match the graphs A, B,

and C in the figure with the functions 𝑓(𝑥), 𝑓′(𝑥) and ∫ 𝑓(𝑡)𝑑𝑡𝑥

0 .

Additional Integral (Briggs Revision Ex.5 #56-61) Evaluate the following integrals.

35) ∫ 𝒔𝒊𝒏 𝟐𝒙

𝟏+𝒄𝒐𝒔𝟐 𝒙 𝒅𝒙 (Hint: 𝒔𝒊𝒏 𝟐𝒙 = 𝟐 𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒙 )

36) ∫𝟏

𝒙𝟐 𝒔𝒊𝒏

𝟏

𝒙 𝒅𝒙 37) ∫

(𝒕𝒂𝒏−𝟏 𝒙)𝟓

𝟏+𝒙𝟐 𝒅𝒙 38)∫

𝒅𝒙

(𝒕𝒂𝒏−𝟏 𝒙)(𝟏+𝒙𝟐)

39) ∫ 𝒔𝒊𝒏−𝟏 𝒙

√𝟏−𝒙𝟐 𝒅𝒙 40) ∫

𝒆𝒙−𝒆−𝒙

𝒆𝒙+𝒆−𝒙 𝒅𝒙