1. f(x,y)= x 3 + x 2 y+x y 2 + y 3 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGaeyypa0JaamiEamaaCaaaleqabaGaaG4maaaakiabgUcaRiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bGaey4kaSIaamiEaiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG5bWaaWbaaSqabeaacaaIZaaaaaaa@483E@

∂f ∂x =3 x 2 +2xy+ y 2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRaamiEaaaacqGH9aqpcaaIZaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWG4bGaamyEaiabgUcaRiaadMhadaahaaWcbeqaaiaaikdaaaaaaa@44C7@

∂f ∂y = x 2 +2xy+3 y 2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOaIyRaamyEaaaacqGH9aqpcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOmaiaadIhacaWG5bGaey4kaSIaaG4maiaadMhadaahaaWcbeqaaiaaikdaaaaaaa@44C8@

∂ 2 f ∂ x 2 = ∂ ∂x ( ∂f ∂x )= ∂ ∂x (3 x 2 +2xy+ y 2 )=6x+2y MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIyRaamiEamaaCaaaleqabaGaaGOmaaaaaaGccqGH9aqpdaWcaaqaaiabgkGi2cqaaiabgkGi2kaadIhaaaGaaiikamaalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kaadIhaaaGaaiykaiabg2da9maalaaabaGaeyOaIylabaGaeyOaIyRaamiEaaaacaGGOaGaaG4maiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaamiEaiaadMhacqGHRaWkcaWG5bWaaWbaaSqabeaacaaIYaaaaOGaaiykaiabg2da9iaaiAdacaWG4bGaey4kaSIaaGOmaiaadMhaaaa@5C44@

∂ 2 f ∂y∂x = ∂ ∂y ( ∂f ∂x )= ∂ ∂y (3 x 2 +2xy+ y 2 )=2x+2y MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIyRaamyEaiabgkGi2kaadIhaaaGaeyypa0ZaaSaaaeaacqGHciITaeaacqGHciITcaWG5baaaiaacIcadaWcaaqaaiabgkGi2kaadAgaaeaacqGHciITcaWG4baaaiaacMcacqGH9aqpdaWcaaqaaiabgkGi2cqaaiabgkGi2kaadMhaaaGaaiikaiaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOmaiaadIhacaWG5bGaey4kaSIaamyEamaaCaaaleqabaGaaGOmaaaakiaacMcacqGH9aqpcaaIYaGaamiEaiabgUcaRiaaikdacaWG5baaaa@5DB3@

∂ 2 f ∂x∂y = ∂ 2 f ∂y∂x =2x+2y MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIyRaamiEaiabgkGi2kaadMhaaaGaeyypa0ZaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIyRaamyEaiabgkGi2kaadIhaaaGaeyypa0JaaGOmaiaadIhacqGHRaWkcaaIYaGaamyEaaaa@4C85@

∂ 2 f ∂ y 2 = ∂ ∂y ( ∂f ∂y )= ∂ ∂y ( x 2 +2xy+3 y 2 )=2x+6y MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIyRaamyEamaaCaaaleqabaGaaGOmaaaaaaGccqGH9aqpdaWcaaqaaiabgkGi2cqaaiabgkGi2kaadMhaaaGaaiikamaalaaabaGaeyOaIyRaamOzaaqaaiabgkGi2kaadMhaaaGaaiykaiabg2da9maalaaabaGaeyOaIylabaGaeyOaIyRaamyEaaaacaGGOaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWG4bGaamyEaiabgUcaRiaaiodacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaaiykaiabg2da9iaaikdacaWG4bGaey4kaSIaaGOnaiaadMhaaaa@5C48@

2. ∫ 1 2 ∫ 1 2 ( x 2 + y 2 ) dx dy MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaWdXbqaaiaacIcacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyEamaaCaaaleqabaGaaGOmaaaakiaacMcacaaMe8UaamizaiaadIhacaaMc8UaamizaiaadMhaaSqaaiaaigdaaeaacaaIYaaaniabgUIiYdaaleaacaaIXaaabaGaaGOmaaqdcqGHRiI8aaaa@4AAF@

= ∫ 1 2 ( x 3 3 + y 2 x ) x=1 x=2  dy MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaqadaqaamaalaaabaGaamiEamaaCaaaleqabaGaaG4maaaaaOqaaiaaiodaaaGaey4kaSIaamyEamaaCaaaleqabaGaaGOmaaaakiaadIhaaiaawIcacaGLPaaaaSqaaiaaigdaaeaacaaIYaaaniabgUIiYdGcdaqhaaWcbaGaamiEaiabg2da9iaaigdaaeaacaWG4bGaeyypa0JaaGOmaaaakiaaysW7caWGKbGaamyEaaaa@4B17@

= ∫ 1 2 [ ( 8 3 − 1 3 )+( 2−1 ) y 2 ]  dy MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaWadaqaamaabmaabaWaaSaaaeaacaaI4aaabaGaaG4maaaacqGHsisldaWcaaqaaiaaigdaaeaacaaIZaaaaaGaayjkaiaawMcaaiabgUcaRmaabmaabaGaaGOmaiabgkHiTiaaigdaaiaawIcacaGLPaaacaWG5bWaaWbaaSqabeaacaaIYaaaaaGccaGLBbGaayzxaaaaleaacaaIXaaabaGaaGOmaaqdcqGHRiI8aOGaaGjbVlaadsgacaWG5baaaa@4B8B@

= ∫ 1 2 ( 7 3 + y 2 ) dy MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaGGOaWaaSaaaeaacaaI3aaabaGaaG4maaaacqGHRaWkcaWG5bWaaWbaaSqabeaacaaIYaaaaOGaaiykaiaaysW7caWGKbGaamyEaaWcbaGaaGymaaqaaiaaikdaa0Gaey4kIipaaaa@42FC@

= ( 7y 3 + y 3 3 ) y=1 y=2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWcaaqaaiaaiEdacaWG5baabaGaaG4maaaacqGHRaWkdaWcaaqaaiaadMhadaahaaWcbeqaaiaaiodaaaaakeaacaaIZaaaaaGaayjkaiaawMcaamaaDaaaleaacaWG5bGaeyypa0JaaGymaaqaaiaadMhacqGH9aqpcaaIYaaaaaaa@4350@

= 7 3 (2−1)+ 1 3 ( 2 3 − 1 3 ) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaI3aaabaGaaG4maaaacaGGOaGaaGOmaiabgkHiTiaaigdacaGGPaGaey4kaSYaaSaaaeaacaaIXaaabaGaaG4maaaacaGGOaGaaGOmamaaCaaaleqabaGaaG4maaaakiabgkHiTiaaigdadaahaaWcbeqaaiaaiodaaaGccaGGPaaaaa@4348@

= 7 3 + 7 3 = 14 3 =4 2 3 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaI3aaabaGaaG4maaaacqGHRaWkdaWcaaqaaiaaiEdaaeaacaaIZaaaaiabg2da9maalaaabaGaaGymaiaaisdaaeaacaaIZaaaaiabg2da9iaaisdadaWcaaqaaiaaikdaaeaacaaIZaaaaaaa@4085@

3. ∫ 0 3 ∫ y 4 (x+y) dx  dy MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaWdXbqaaiaacIcacaWG4bGaey4kaSIaamyEaiaacMcacaaMe8UaamizaiaadIhaaSqaaiaadMhaaeaacaaI0aaaniabgUIiYdaaleaacaaIWaaabaGaaG4maaqdcqGHRiI8aOGaaGPaVlaadsgacaWG5baaaa@4918@

= ∫ 0 3 ( x 2 2 +yx ) x=y x=4  dy MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaqadaqaamaalaaabaGaamiEamaaCaaaleqabaGaaGOmaaaaaOqaaiaaikdaaaGaey4kaSIaamyEaiaadIhaaiaawIcacaGLPaaadaqhaaWcbaGaamiEaiabg2da9iaadMhaaeaacaWG4bGaeyypa0JaaGinaaaaaeaacaaIWaaabaGaaG4maaqdcqGHRiI8aOGaaGjbVlaadsgacaWG5baaaa@4A52@

= ∫ 0 3 [ ( 16 2 − y 2 2 )+y( 4−y ) ]  dy MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaWadaqaamaabmaabaWaaSaaaeaacaaIXaGaaGOnaaqaaiaaikdaaaGaeyOeI0YaaSaaaeaacaWG5bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaaaaaiaawIcacaGLPaaacqGHRaWkcaWG5bWaaeWaaeaacaaI0aGaeyOeI0IaamyEaaGaayjkaiaawMcaaaGaay5waiaaw2faaaWcbaGaaGimaaqaaiaaiodaa0Gaey4kIipakiaaysW7caWGKbGaamyEaaaa@4CCA@

= ∫ 0 3 ( 8− y 2 2 +4y− y 2 )  dy MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaqadaqaaiaaiIdacqGHsisldaWcaaqaaiaadMhadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaaaaiabgUcaRiaaisdacaWG5bGaeyOeI0IaamyEamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaWcbaGaaGimaaqaaiaaiodaa0Gaey4kIipakiaaysW7caWGKbGaamyEaaaa@48BD@

= ∫ 0 3 ( 8+4y− 3 y 2 2 ) dy MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaqadaqaaiaaiIdacqGHRaWkcaaI0aGaamyEaiabgkHiTmaalaaabaGaaG4maiaadMhadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaaaaaGaayjkaiaawMcaaiaaysW7caWGKbGaamyEaaWcbaGaaGimaaqaaiaaiodaa0Gaey4kIipaaaa@4692@

= ( 8y+2 y 2 − y 3 2 ) y=0 y=3 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaI4aGaamyEaiabgUcaRiaaikdacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0YaaSaaaeaacaWG5bWaaWbaaSqabeaacaaIZaaaaaGcbaGaaGOmaaaaaiaawIcacaGLPaaadaqhaaWcbaGaamyEaiabg2da9iaaicdaaeaacaWG5bGaeyypa0JaaG4maaaaaaa@461D@

= 8(3−0)+2(9−0)− 1 2 (27−0) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaacIcacaaIZaGaeyOeI0IaaGimaiaacMcacqGHRaWkcaaIYaGaaiikaiaaiMdacqGHsislcaaIWaGaaiykaiabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaaiikaiaaikdacaaI3aGaeyOeI0IaaGimaiaacMcaaaa@46BF@

= 24+18− 27 2 =42− 27 2 =28 1 2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaisdacqGHRaWkcaaIXaGaaGioaiabgkHiTmaalaaabaGaaGOmaiaaiEdaaeaacaaIYaaaaiabg2da9iaaisdacaaIYaGaeyOeI0YaaSaaaeaacaaIYaGaaG4naaqaaiaaikdaaaGaeyypa0JaaGOmaiaaiIdadaWcaaqaaiaaigdaaeaacaaIYaaaaaaa@46BE@

4. R is the region bounded below by the graph of y= x 2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2da9iaadIhadaahaaWcbeqaaiaaikdaaaaaaa@39D8@  and above by the graph of the line y = 1. These two graphs intersect at the points (-1,1) and (1,1). To find that simply solve the equation x 2 =1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabg2da9iaaigdaaaa@399F@  to get x = -1 and x = 1.

So, ∬ R (1+x) dx dy MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8GuaeaacaGGOaGaaGymaiabgUcaRiaadIhacaGGPaGaaGjbVlaadsgacaWG4bGaaGPaVlaadsgacaWG5baaleaacaWGsbaabeqdcqGHRiI8cqGHRiI8aaaa@45B2@

= ∫ −1 1 ∫ x 2 1 (1+x) dy dx MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaWdXbqaaiaacIcacaaIXaGaey4kaSIaamiEaiaacMcacaaMe8UaamizaiaadMhacaaMc8UaamizaiaadIhaaSqaaiaadIhadaahaaadbeqaaiaaikdaaaaaleaacaaIXaaaniabgUIiYdaaleaacqGHsislcaaIXaaabaGaaGymaaqdcqGHRiI8aaaa@4AA8@

= ∫ −1 1 (y+xy) y= x 2 y=1  dx MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaGGOaGaamyEaiabgUcaRiaadIhacaWG5bGaaiykamaaDaaaleaacaWG5bGaeyypa0JaamiEamaaCaaameqabaGaaGOmaaaaaSqaaiaadMhacqGH9aqpcaaIXaaaaaqaaiabgkHiTiaaigdaaeaacaaIXaaaniabgUIiYdGccaaMe8UaamizaiaadIhaaaa@4A42@

= ∫ −1 1 [ (1− x 2 )+x(1− x 2 ) ]  dx MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaadaWadaqaaiaacIcacaaIXaGaeyOeI0IaamiEamaaCaaaleqabaGaaGOmaaaakiaacMcacqGHRaWkcaWG4bGaaiikaiaaigdacqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaaiykaaGaay5waiaaw2faaaWcbaGaeyOeI0IaaGymaaqaaiaaigdaa0Gaey4kIipakiaaysW7caWGKbGaamiEaaaa@4BEA@

= ∫ −1 1 (1− x 2 +x− x 3 ) dx MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaGGOaGaaGymaiabgkHiTiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG4bGaeyOeI0IaamiEamaaCaaaleqabaGaaG4maaaakiaacMcacaaMe8UaamizaiaadIhaaSqaaiabgkHiTiaaigdaaeaacaaIXaaaniabgUIiYdaaaa@47DB@

= ( x− x 3 3 + x 2 2 − x 4 4 ) x=−1 x=1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWG4bGaeyOeI0YaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIZaaaaaGcbaGaaG4maaaacqGHRaWkdaWcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaaaaiabgkHiTmaalaaabaGaamiEamaaCaaaleqabaGaaGinaaaaaOqaaiaaisdaaaaacaGLOaGaayzkaaWaa0baaSqaaiaadIhacqGH9aqpcqGHsislcaaIXaaabaGaamiEaiabg2da9iaaigdaaaaaaa@4A00@

= (1−(−1))− 1 3 (1−(−1))+ 1 2 (1−1)− 1 4 (1−1) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHsislcaGGOaGaeyOeI0IaaGymaiaacMcacaGGPaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaG4maaaacaGGOaGaaGymaiabgkHiTiaacIcacqGHsislcaaIXaGaaiykaiaacMcacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaiaacIcacaaIXaGaeyOeI0IaaGymaiaacMcacqGHsisldaWcaaqaaiaaigdaaeaacaaI0aaaaiaacIcacaaIXaGaeyOeI0IaaGymaiaacMcaaaa@50BE@

= 2− 2 3 =1 1 3 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgkHiTmaalaaabaGaaGOmaaqaaiaaiodaaaGaeyypa0JaaGymamaalaaabaGaaGymaaqaaiaaiodaaaaaaa@3C69@