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∫ (−? 2 + 9)?? = −? 3 3 3 0 + 9? → [ −(3) 3 3 + 9(3)] − [ −(0) 3 3 + 9(0)] = 18 ∫ (? 2 − 9)?? = ? 3 3 4 3 − 9? → [ (4) 3 3 − 9(4)] − [ (3) 3 3 − 9(3)] = 10 3 |? 2 − 9| ?? = 18 + 10 3 = 64 3 4 0 9) ∫ (1 + ???? )?? = ? − ???? → [? − ????] ? 0 − [0 − ???0] = 2 + ? 10) ∫ ( 1 − ??? 2 ? ??? 2 ? ) ?? = ? 4 0 ∫ ( ??? 2 ? ??? 2 ? ) ?? = ∫ 1 ?? ? 4 0 ? 4 0 = ? → ? 4 − 0 = ? 4
Calculus Page 89 of 136 11) ∫ (??? 2 ?)?? = ???? → [??? ? 6 ] ? 6 −? 6 − [??? −? 6 ] = 2√3 3 12) ∫ (2 − ??? 2 ?)?? ? 2 ? 4 = 2? + ???? → [2( ? 2 ) + ??? ? 2 ] − [2 ( ? 4 ) + ??? ? 4 ] = ? − 2 2 13) ∫ (4???𝜃 ???𝜃)?𝜃 = 4???𝜃 → [4??? ? 3 ] − [4??? −? 3 ] = 0 ? 3 −? 3 14) ∫ ( ??? 2 𝜃 ??? 2 𝜃 + 1 ) ?𝜃 = ? 4 0 ( ??? 2 𝜃 ??? 2 𝜃 ) ?𝜃 = ∫ ?𝜃 = 𝜃 → ? 4 ? 4 0 ? 4 0 − 0 = ? 4 Question 2: Find the area of the region bounded by the graphs of the equations 1) ? = 5? 2 + 2 , ? = 0 , ? = 2 , ? = 0 5? 2 + 2 = 0 → ?? ???????? ???? = ∫ (5? 2 + 2) 2 0 − (0)?? = 5? 3 3 + 2? → [ 5(2) 3 3 + 2(2)] − [ 5(0) 3 3 + 2(0)] = 52 3 2) ? = 1 + ? 3 , ? = 0 , ? = 8 , ? = 0 1 + ? 3 = 0 → ? 3 = −1 → ? = −1 ???????? ???? = ∫ (1 + ? 3 ) 8 0 − (0)?? = ? + 3? 4 3 4 → [(8) + 3(8) 4 3 4 ] − [(0) + 3(0) 4 3 4 ] = 20 3) ? = −? 2 + 4? , ? = 0 −? 2 + 4? = 0 → ? = 0, ? = 4 ???? = ∫ (−? 2 + 4?) 4 0 − (0)?? = −? 3 3 + 2? 2 → [ −(4) 3 3 + 2(4) 2 ] − [ −(0) 3 3 + 2(0) 2 ] = 32 3 4) ? = 1 − ? 4 , ? = 0 1 − ? 4 = 0 → ? = ±1 ???? = ∫ (1 − ? 4 ) 1 −1 − (0)?? = ? − ? 5 5 → [(1) − (1) 5 5 ] − [(−1) − (−1) 5 5 ] = 8 5 Question 3:
Calculus Page 90 of 136 Find the value of ? guaranteed by the mean value theorem for integrals for the function over the given interval 1) ?(?) = ? 3 , [0, 3] ∫ ? 3 ?? = ? 4 4 3 0 → [ (3) 4 4 ] − [ (0) 4 4 ] = 81 4 ?(?)(? − ?) = ∫ ?(?)?? ? ? ? 3 (3 − 0) = 81 4 → 3? 3 = 81 4 → 12? 3 − 81 = 0 → ? ≈ 1.8899 2) ?(?) = ? 2 4 , [0, 6] ? 2 4 ?? = ? 3 12 6 0 → [ (6) 3 12 ] − [ (0) 3 12 ] = 18 ?(?)(? − ?) = ∫ ?(?)?? ? ? ? 2 4 (6 − 0) = 18 → 6? 2 4 = 18 → 6? 2 = 72 → ? = 2√3 3) ?(?) = 2??? 2 ?, [ −? 4 , ? 4 ] 2??? 2 ??? = 2???? ? 4 −? 4 → [2??? ? 4 ] − [2??? −? 4 ] = 4 ?(?)(? − ?) = ∫ ?(?)?? ? ? 2??? 2 ? ( ? 4 + ? 4 ) = 4 → 2??? 2 ? ( ? 2 ) = 4 → ???? 2 ? = 4 → ??? 2 ? = 4 ? → ??? 2 ? = ? 4 → ? ≈ 0.4817 Question 4: Find the average value of the function over the given interval and all values of ? in the given interval for which the function is equal its average value 1)? (?) = 9 − ? 2 , [−3, 3] Average value = 1 3—3 (9 − ? 2 )?? = 1 6 3 –3 (9? − ? 3 3 ) 1 6 ([9(3) − (3) 3 3 ] − [9(−3) − (−3) 3 3 ]) = 6 9 − ? 2 = 6 → ? = ±√3 2) ?(?) = ? 3 , [0, 1] Average value = 1 1−0 ∫ ? 3 ?? = 1 1 1 0 ( ? 4 4 ) → 1 1 ([ (1) 4 4 ] − [ (0) 4 4 ]) = 1 4 ? 3 = 1 4 → 4? 3 = 1 → 4? 3 − 1 = 0 → ? ≈ 0.63 3)?(?) = ????, [0, ?] Average value = 1 ?−0 ?????? = 1 ? ? 0 (−????) → 1 ? ([−????] − [−???0]) = 2 ? ???? = 2 ? → ? ≈ 0.690 , ? ≈ 2.451 Question 5
Calculus Page 91 of 136 A particle is moving along a line so that its velocity ?(?) = ? 3 − 10? 2 + 29 ? − 20 1. what is the displacement of the particle on the time interval 1 ≤ ? ≤ 5 ?(?) = ∫ (? 3 − 10? 2 + 29 ? − 20)?? = ? 4 4 5 1 10? 3 3 + 29? 2 2 − 20? [ (5) 4 4 10(5) 3 3 + 29(5) 2 2 − 20(5)] − [ (1) 4 4 10(1) 3 3 + 29(1) 2 2 − 20(1)] = 32 3 2. what is the total distance traveled by the particle on the time interval 1 ≤ ? ≤ 5 ? 3 − 10? 2 + 29 ? − 20 = 0 → ? = 1, ? = 4, ? = 5 ????? ???????? = ∫ (? 3 − 10? 2 + 29 ? − 20)?? − ∫ (? 3 − 10? 2 + 29 ? − 20)?? 5 4 4 1 [ (4) 4 4 10(4) 3 3 + 29(4) 2 2 − 20(4)] − [ (1) 4 4 10(1) 3 3 + 29(1) 2 2 − 20(1)] − [ (5) 4 4 10(5) 3 3 + 29(5) 2 2 − 20(5)] − [ (4) 4 4 10(4) 3 3 + 29(4) 2 2 − 20(1)] = 71 6 Question 6 Find the derivative of 1. ?(?) = ∫ ???? ?? ? 3 𝜋 2 ? (?) = (???? 3 )(3? 2 ) 2. ?(?) = ∫ ???𝜃 2 ?𝜃 ? 2 0 ? (?) = (???? 4 )(2?) 3. ?(?) = ∫ ? ?? ?𝑖?? 0 ? (?) = (√???? )(????) 4. ?(?) = ∫ 1 ? 3 ?? ? 2 2 ? (?) = ( 1 ? 6 ) (2?) = 2 ? 5 5. ?(?) = ∫ ??? 3 ? ?? ? 2 0 ? (?) = (??? 3 ? 2 )(2?) Subtopic 4.6 (integration by substitution)
Calculus Page 92 of 136 Question 1: Find the indefinite integral 1) ∫(? 2 − 9) 3 (2?) ?? ? = ? 2 − 9 ?? = 2? ?? ?? = ?? 2? ∫(? 2 − 9) 3 (2?) ?? = ∫ ? 3 (2?)( ?? 2? ) = ∫ ? 3 ?? = ? 4 4 + ? → (? 2 − 9) 4 4 + ? 2) ∫ 3 − 4? 2 3 (−8?)?? ? = 3 − 4? 2 ?? = −8? ?? ?? = ?? −8? 3 − 4? 2 3 (−8?)?? = ∫ ? 3 (−8?) ( ?? −8? ) = ∫ ? 3 ?? = 3 4 ? 4 3 + ? → 3 4 (3 − 4? 2 ) 4 3 + ? 3) ∫? 3 (? 4 + 3) 2 ?? ? = ? 4 + 3 ?? = 4? 3 ?? = ?? 4? 3 ∫ ? 3 (? 4 + 3) 2 ?? = ∫ ? 3 (?) 2 ?? 4? 3 = ∫ ? 2 4 ?? = ? 3 12 + ? → (? 4 + 3) 3 12 + ? 4) ∫? 3 2? 4 + 3 ?? ? = 2? 4 + 3 ?? = 8? 3 ?? = ?? 8? 3 ∫ ? 3 2? 4 + 3 ?? = ∫ ? 3 ? ?? 8? 3 = ∫ ? 1 2 8 ?? = 2? 3 2 24 + ? → (2? 4 + 3) 3 2 12 + ? 5) ∫5? 1 − ? 2 3 ??
Calculus Page 93 of 136 ? = 1 − ? 2 → ? 2 = 1 − ? ?? = −2? ?? ?? = ?? −2? ∫ 5? 1 − ? 2 3 ?? = ∫ 5? 3 ? ?? −2? = ∫ 5? 2 ? −2 ?? = ∫ 5(1 − ?) ? −2 ?? 5? 1 2 −2 ?? − ∫ 5?