Lab3: Symbolic integration and the fundamental theorem of calculus - going backwards Jeremy Johnson In the previous lab, Maple was used to explore the concept of definite integration. The area under a curve was approximated using the sum of the areas of boxes, and definite integration was viewed as a limiting process, where thinner and thinner boxes were used. For example, consider the function 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. The area under the curve 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as 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goes from 0 to 1 can be estimated by summing up the area of 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boxes where the dimensions of the 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-th box is 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2OVEiaUYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUqdW5kZXJsaW5lR0Y3LyUqc3Vic2NyaXB0R0Y3LyUsc3VwZXJzY3JpcHRHRjcvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdGQy8lJ29wYXF1ZUdGNy8lK2V4ZWN1dGFibGVHRjcvJSlyZWFkb25seUdGNy8lKWNvbXBvc2VkR0Y3LyUqY29udmVydGVkR0Y3LyUraW1zZWxlY3RlZEdGNy8lLHBsYWNlaG9sZGVyR0Y3LyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUqbWF0aGNvbG9yR0ZDLyUvbWF0aGJhY2tncm91bmRHRkMvJStmb250ZmFtaWx5R0YxLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lKW1hdGhzaXplR0Y0LUkjbW9HRiQ2M1EiPUYnLyUlZm9ybUdRJmluZml4RicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRmlvLyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShtYXhzaXplR1EpaW5maW5pdHlGJy8lKG1pbnNpemVHUSIxRicvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lMGZvbnRfc3R5bGVfbmFtZUdGVi8lJXNpemVHRjQvJStmb3JlZ3JvdW5kR0ZDLyUrYmFja2dyb3VuZEdGQy1JI21uR0YkNjlGZXBGL0YyRjUvRjlGN0Y7Rj1GP0ZBRkRGRkZIRkpGTEZORlBGUkZURldGWUZlbi9GaG5RJ25vcm1hbEYnRmpuLUZdbzYzUSMuLkYnL0Zhb1EocG9zdGZpeEYnRmNvRmVvL0Zob1EwbWVkaXVtbWF0aHNwYWNlRicvRltwUSQwZW1GJ0ZccEZecEZgcEZjcEZmcEZocEZqcEZccUZecUZgcUZicS1GLDY5USJORidGL0YyRjVGOEY7Rj1GP0ZBRkRGRkZIRkpGTEZORlBGUkZURldGWUZlbkZnbkZqbi1GXW82M1EiLkYnRmBvRmNvRmVvL0Zob0ZickZhckZccEZecEZgcEZjcEZmcEZocEZqcEZccUZecUZgcUZicQ== The following figure illustrates this when 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 Ni0tSSlQT0xZR09OU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkNyY3JCQiIiFGLUYsNyQkIisrKytdNyEjNUYsNyRGLyQiKysrXWk6ISM2NyRGLEYzLUktVFJBTlNQQVJFTkNZR0YoNiMkIiIiRi0tRiQ2JDcmRi43JCQiKysrKytERjFGLDckRkAkIisrKytdaUY1NyRGL0ZDRjctRiQ2JDcmRj83JCQiKysrK11QRjFGLDckRkokIisrK0QxOUYxNyRGQEZNRjctRiQ2JDcmRkk3JCQiKysrKytdRjFGLDckRlRGQDckRkpGQEY3LUYkNiQ3JkZTNyQkRkRGMUYsNyRGZm4kIisrK0QxUkYxNyRGVEZobkY3LUYkNiQ3JkZlbjckJCIrKysrK3ZGMUYsNyRGX28kIisrKytEY0YxNyRGZm5GYm9GNy1GJDYkNyZGXm83JCQiKysrK10oKUYxRiw3JEZpbyQiKysrRGN3RjE3JEZfb0ZccEY3LUYkNiQ3JkZobzckRjpGLDckRjpGOjckRmlvRjpGNy1JJ0NVUlZFU0dGKDYkN1NGKzckJCIzZW1tbTthcnpAISM+JCIzISpSVmwoSGY2diUhI0A3JCQiM1tMTCRlOXVpMiVGXHEkIjNJRiopPiI0LDttIiEjPzckJCIzbm1tbSJ6XyI0aUZccSQiMylmPkUhUnlOYlFGZXE3JCQiM1ttbW1UJnBoTilGXHEkIjMpNHgyNCVwYiMpcEZlcTckJCIzQ0xMZSo9KUhcNSEjPSQiM25nR20hcEU1NSJGXHE3JCQiM2dtbSJ6LzN1QyJGY3IkIjN1b3MrUW8tYzpGXHE3JCQiMyUpKioqXDdMUkRYIkZjciQiM3UnPkcpMzAoKTRARlxxNyQkIjNdbW0ielInb2s7RmNyJCIzM20oM00hMz1yRkZccTckJCIzdyoqKlxpNWBoKD1GY3IkIjN6QTo0eS8mKj5ORlxxNyQkIjNXTExMM0VuJDQjRmNyJCIzeS5gKTMqXFkkUSVGXHE3JCQiM3FtbTsvUkUmRyNGY3IkIjMnZTIobzY2VkFfRlxxNyQkIjMiKSoqKioqXEtdNF0jRmNyJCIzdS93NkdEdmFpRlxxNyQkIjMkKioqKioqXFBBdnIjRmNyJCIzITRrRCdleSNcUShGXHE3JCQiMykqKioqKipcbkhpI0hGY3IkIjNDMDEmMzY/R2MpRlxxNyQkIjNqbW0ieipldjpKRmNyJCIzOiVcLWEiWyR6cSpGXHE3JCQiMz9MTEwzNDdUTEZjciQiM3Y6LVQjKikzajYiRmNyNyQkIjMsTExMTFkuS05GY3IkIjM/OG01bG9fWjdGY3I3JCQiM3cqKipcN283VHYkRmNyJCIzSTFBTi1pTDQ5RmNyNyQkIjMnR0xMTFEqb11SRmNyJCIzIzRGZS5tJXpnOkZjcjckJCIzQSsrRCI9bGo7JUZjciQiM3pNS04jKSlmZXQiRmNyNyQkIjMxKyt2ViZSPFAlRmNyJCIzeFgoUVFtNTciPkZjcjckJCIzV0xMJGU5RWdlJUZjciQiMyNRcEU1ZWpKNSNGY3I3JCQiM0dMTGVSIjNHeSVGY3IkIjNdZWsrcWBfKEcjRmNyNyQkIjNjbW07L1QxJipcRmNyJCIzR01NWlNsMSZcI0ZjcjckJCIzJmVtO3pSUWJAJkZjciQiM1I5Vix5Uz0/RkZjcjckJCIzXCoqKlwoPT5ZMmFGY3IkIjNdVUpGU1cxQ0hGY3I3JCQiMzltbTt6WHU5Y0ZjciQiM1NZIkcqb2NgX0pGY3I3JCQiM2wqKioqKipceSkpR2VGY3IkIjMnPncoeWNMZihSJEZjcjckJCIzJyopKioqXGlfUVFnRmNyJCIzIXpyUHlsNGlrJEZjcjckJCIzQCoqKlw3eSUzVGlGY3IkIjMjXC52WSNSNiYqUUZjcjckJCIzNSoqKipcUCFbaFknRmNyJCIzIyozbEdXcTUiPSVGY3I3JCQiM2tLTEwkUXgkb21GY3IkIjM7XV5sI3BEblclRmNyNyQkIjMhKSoqKioqXFArVilvRmNyJCIzbThES2wiZiRSWkZjcjckJCIzP21tInpwZSp6cUZjciQiM2UmZT9rXiJlN11GY3I3JCQiMyUpKioqKipcI1wnUUgoRmNyJCIzeUNYVGFsLz9gRmNyNyQkIjNHS0xlOVM4JlwoRmNyJCIzWzBrbCpRLnhoJkZjcjckJCIzUioqKlxpPz1icShGY3IkIjNfP11vIzMsdiRmRmNyNyQkIjMiSExMJDNzPzZ6RmNyJCIzZ0cmPSRcKj4oZWlGY3I3JCQiM2EqKipcN2BXbDcpRmNyJCIzSUYqUT1nc1NnJ0ZjcjckJCIzI3BtbW0nKlJSTClGY3IkIjMzKy8hb2BiYSVwRmNyNyQkIjNRbW07YTwuWSYpRmNyJCIzJSopXEFWKGVZLnRGY3I3JCQiMz1MTGU5dE9jKClGY3IkIjNzTC56YW9SbndGY3I3JCQiM3UqKioqKipcUWtcKilGY3IkIjNSIkclPS9EaDQhKUZjcjckJCIzQ0xMJDNkZzY8KkZjciQiM3Mpeis7aT01VClGY3I3JCQiM0ltbW1teEdwJCpGY3IkIjN1azRZRGBOeSgpRmNyNyQkIjNBKytEIm9LMGUqRmNyJCIzK0sqXGNrZyd5IipGY3I3JCQiM0ErK3Y9NXMjeSpGY3IkIjMjUUlwSTBqLGQqRmNyRmNwLUknQ09MT1VSR0YoNiZJJFJHQkdGKCQiIzUhIiJGLEYsLUkrQVhFU0xBQkVMU0dGJTYkUSJ4RihRIUYoLUklVklFV0dGJTYkO0YsRjpJKERFRkFVTFRHRig= The definite integral (area under the cuve 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where 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runs from 0 to 1) was computed by taking the limit as 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goes to infinity. 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 Recall that indefinite integration is the problem of anti-differentiation. I.E. given a function 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find a function 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whose derivative is 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the indefinite integral 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. The fundamental theorem of calculus relates this process of anti-differentiaiton to definite integration. It allows algebraic techniques, which will be explored in this lab, to be used to get the results obtained through this limiting process without having to explicitly compute limits. In the example above, 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and we observe that 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More generally, the fundamental theorem of calculus states that 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR0ZDLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSptYXRoY29sb3JHRkMvJS9tYXRoYmFja2dyb3VuZEdGQy8lK2ZvbnRmYW1pbHlHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUpbWF0aHNpemVHRjQtRiM2JUYrLUYjNidGKy1GIzYnRistSSZtZnJhY0dGJDYqLUYjNiMtSSNtb0dGJDYzUTAmRGlmZmVyZW50aWFsRDtGJy8lJWZvcm1HUSdwcmVmaXhGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRmRwLyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShtYXhzaXplR1EpaW5maW5pdHlGJy8lKG1pbnNpemVHUSIxRicvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lMGZvbnRfc3R5bGVfbmFtZUdGVi8lJXNpemVHRjQvJStmb3JlZ3JvdW5kR0ZDLyUrYmFja2dyb3VuZEdGQy1GIzYlRistRiM2JEZnby1GLDY5USJ4RidGL0YyRjVGOEY7Rj1GP0ZBRkRGRkZIRkpGTEZORlBGUkZURldGWUZlbkZnbkZqbkYrLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zbcy8lKWJldmVsbGVkR0Y3RltyRl1yLUZobzYzUTEmSW52aXNpYmxlVGltZXM7RicvRlxwUSZpbmZpeEYnRl5wRmBwRmJwRmVwRmdwRmlwRltxRl5xRmFxRmNxRmVxRmdxRmlxRltyRl1yLUYjNiYtRiw2OVEiRkYnRi9GMkY1RjhGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GZ25Gam4tRmhvNjNRMCZBcHBseUZ1bmN0aW9uO0YnRmNzRl5wRmBwRmJwRmVwRmdwRmlwRltxRl5xRmFxRmNxRmVxRmdxRmlxRltyRl1yLUYjNiUtRmhvNjNRIihGJ0ZbcC9GX3BGOkZgcC9GY3BRLnRoaW5tYXRoc3BhY2VGJy9GZnBGZHQvRmhwRjpGaXBGW3FGXnFGYXFGY3FGZXFGZ3FGaXFGW3JGXXItRiM2I0Zjci1GaG82M1EiKUYnL0ZccFEocG9zdGZpeEYnRmJ0RmBwRmN0L0ZmcFEydmVyeXRoaW5tYXRoc3BhY2VGJ0ZmdEZpcEZbcUZecUZhcUZjcUZlcUZncUZpcUZbckZdckYrRistRmhvNjNRIj1GJ0Zjc0ZecEZgcC9GY3BRL3RoaWNrbWF0aHNwYWNlRicvRmZwRmR1RmdwRmlwRltxRl5xRmFxRmNxRmVxRmdxRmlxRltyRl1yLUYjNiYtRiw2OVEiZkYnRi9GMkY1RjhGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GZ25Gam5GanNGXXRGK0YrRitGKw==. For example, when 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 and a=0 and b=1, we saw that 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The function 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 is such that