[H2 MATH] Chapter 11 - Applications of Integration

SHALYN TAY COPYRIGHTED © A LEVEL H2 MATHEMATICSAPPLICATIONS OF INTEGRATION

CHAPTERANALYSIS MASTERY EXAM WEIGHTAGE SHALYN TAY COPYRIGHTED ©

Approximation By Rectangles SHALYN TAY COPYRIGHTED © Overestimation Underestimation Use Summation to Approximate Area x y 1 𝑛 2 𝑛 3 𝑛 𝑛 𝑛=1… 𝒚=𝟐𝒙𝟐 2 𝑛2 Height of rectangle 2𝑛 𝑛 2 𝐀𝐩𝐩𝐫𝐨𝐱𝐢𝐦𝐚𝐭𝐞𝐝𝐀𝐫𝐞𝐚 =1 n 2 𝑛2+1 𝑛 8 𝑛2+⋯+1 𝑛2𝑛 𝑛 2 =2 𝑛3෍ 𝑟=1 𝑛 𝑟2=2 𝑛3×𝑛ሺ𝑛+1ሻሺ2𝑛+1ሻ 6 =ሺ𝑛+1ሻሺ2𝑛+1ሻ 2𝑛2 =2𝑛2+3𝑛+1 2𝑛2 =1+3 2𝑛+1 2𝑛2 As 𝑛→∞, 3 2𝑛→0and 1 2𝑛2→0 Therefore, 𝐴𝑟𝑒𝑎→1from 𝑥=0to 𝑥=1

SHALYN TAY COPYRIGHTED © Evaluating Area Bound By Curves & Axes x y x y 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝐴𝑟𝑒𝑎 =න௕ ௖ 𝑓𝑥𝑑𝑥 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒𝐴𝑟𝑒𝑎 =−න𝑎 ௕ 𝑓𝑥𝑑𝑥=න𝑎 ௕ 𝑓ሺ𝑥ሻ𝑑𝑥 X-AXIS Y-AXIS 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝐴𝑟𝑒𝑎 =න𝑎 ௕ 𝑓𝑦𝑑𝑦 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒𝐴𝑟𝑒𝑎 =−න𝑎 ௕ 𝑓𝑦𝑑𝑦=න𝑎 ௕ 𝑓ሺ𝑦ሻ𝑑𝑦b c c b

SHALYN TAY COPYRIGHTED © Evaluating Area Bound Between Curves x y X-AXIS Y-AXIS x y x y x y x y x y 𝐴𝑟𝑒𝑎 =න𝑎 ௕ 𝑓1𝑥−𝑓2ሺ𝑥ሻ𝑑𝑥 𝐴𝑟𝑒𝑎 =න௖ 𝑑 𝑓1𝑦−𝑓2ሺ𝑦ሻ𝑑𝑦 = = - -

SHALYN TAY COPYRIGHTED © Volume Of Solid Of Revolution Rotation About X-AXIS Rotation About Y-AXIS x y x y x y x y 𝑉𝑜𝑙𝑢𝑚𝑒 =𝜋න𝑎 ௕ 𝑓1𝑥2−𝑓2𝑥2𝑑𝑥 𝑉𝑜𝑙𝑢𝑚𝑒 =𝜋න𝑎 ௕ 𝑓𝑥2𝑑𝑥 𝑉𝑜𝑙𝑢𝑚𝑒 =𝜋න௖ 𝑑 𝑓1𝑦2−𝑓2𝑦2𝑑𝑦 𝑉𝑜𝑙𝑢𝑚𝑒 =𝜋න௖ 𝑑 𝑓𝑦2𝑑𝑦

7 For more notes & learning materials,visit:www.overmugged.com ‘A͛ levels crash course program IG handle:@overmugged DARRELL ER (COPYRIGHTED) © Join our telegram channel:@overmuggedAlevels Need help? ShalynTay(Private tutor with 4 yearsof experience) 82014166(Whatsapp) @shalyntay(telegram username) Professionally designed crash courseto help you get a condensed revision before your ‘A’ Levels! Each H2 subject will have 3 crash course moduleswhich will cover their entire H2 syllabus. The 4 hour module focuses on going through key concepts andidentifying commonly tested questions! The crash courses modules will begin in June 2021 and last till Oct 2021. Pre-register now on our websiteand secure your slots!