MAT1505-01/02 21F Quiz 4 Print Name (Last, First) __________________ , _______________|____
Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use EQUAL SIGNS and arrows when appropriate. Always SIMPLIFY expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation). INDICATE where technology is used and what type (Maple, GC).1. a) Use the relation 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 (cheap trick polynomial long division!) to complete the integration by parts of 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, showing all steps.b) Use that result to evaluate the area between the two graphs LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ4RidGL0YyLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVi1GLDYlUSdhcmN0YW5GJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2JS1JJm1mcmFjR0YkNihGTy1GIzYlLUYsNiVRImFGJ0YvRjJGL0YyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zoby8lKWJldmVsbGVkR0Y9LyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZVL0ZOUSwwLjMzMzMzMzNlbUYnRitGNUZPRlJGWC1GZ242JC1GIzYlLUkjbW5HRiQ2JEZlb0Y5Rl1wRjlGOUZdcEY5 by doing an obvious variable substitution first to the area integral and then using the result from part a) for the antiderivative. Support your initial integral formula for the area with an explanatory diagram. Does your result make sense given your plot? Can you compare with the rectangle containing the two curves?c) What is the average value of 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 over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUZLNiVRImFGJ0ZORlEvJStleGVjdXRhYmxlR0Y4Ri8=?2. Evaluate 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 using integration by parts, step by step.solutionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIihGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdRJmZhbHNlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSZwbG90c0YnRi9GMi1GNjYtUSI6RidGOS9GPEY/Rj0vRkFGP0ZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlgvJStleGVjdXRhYmxlR0Y/Rjk=1.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R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choice: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 have a few variations due to trig identities. In the integral of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjZHZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic= we do a variable substitution 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
We can pick LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY8LUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ4RidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEjZHZGJ0YvRjJGNS1GLDYlUSRzaW5GJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2JUZPLyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZWL0ZORlctRiw2JVEkY29zRidGam5GOUZbb0Ziby1GLDYlUSNkeEYnRi9GMkZSLUYsNiVRInZGJ0YvRjJGNS1JJm1mcmFjR0YkNigtRiM2Ji1JJW1zdXBHRiQ2JUZnbi1GIzYlLUkjbW5HRiQ2JFEiMkYnRjlGL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Zbb0Zgb0Y5RmdwLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZlcS8lKWJldmVsbGVkR0Y9LUY2Ni5GZG8vJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjlGO0Y+RkBGQkZERkZGSEZWRmVvLUY2NjBRI29yRicvJSVib2xkR0YxRlxyL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0ZlckY7Rj5GQEZCRkRGRkZIRlZGZW9GanEtRiw2JkZecEYvRlxyRjItRjY2LkY4RlxyRjlGO0Y+RkBGQkZERkZGSEZKRk0tRjY2LlEqJnVtaW51czA7RidGXHJGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmBzLUZgcDYoLUYjNiUtRmVwNiUtRiw2JkZob0ZqbkZcckY5LUYjNiQtRmpwNiVGXHFGXHJGOUY5Rl1xLUZcbzYlLUYjNiQtRiw2JkZRRi9GXHJGMkY5RlxyRjlGOUZqc0ZgcUZjcUZmcUZocUZgb0Y5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbW9HRiQ2LVEiJ0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR1EmMC4wZW1GJy1GLDYtUSJ+RidGL0YyRjVGN0Y5RjtGPUY/L0ZCRkZGRC1GLDYtUSsmSW50ZWdyYWw7RidGL0YyRjUvRjhRJXRydWVGJ0Y5L0Y8Rk9GPUY/RkpGRC1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dGTy9GMFEnaXRhbGljRidGRy1GUjYlUSRzaW5GJy9GVkY0Ri8tSShtZmVuY2VkR0YkNiQtRiM2JUZRLyUrZXhlY3V0YWJsZUdGNEYvRi9GRy1GUjYlUSRjb3NGJ0ZmbkYvRmduRkctRiw2LVEwJkRpZmZlcmVudGlhbEQ7RidGLy9GM1EmdW5zZXRGJy9GNkZlby9GOEZlby9GOkZlby9GPEZlby9GPkZlby9GQEZlb0ZKRkRGUUYrLUYsNi1RIj1GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGYHBGR0ZHLUkmbWZyYWNHRiQ2KC1GIzYoRlFGRy1JJW1zdXBHRiQ2JUZZLUYjNiUtSSNtbkdGJDYkUSIyRidGL0ZVRlcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRmduRlxvRi9GanAvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmhxLyUpYmV2ZWxsZWRHRjQtRiw2LVEoJm1pbnVzO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZhckZLLUZjcDYoLUYjNiZGZ3BGZ25GXG9GL0ZqcEZjcUZmcUZpcUZbckZHRmFvRlEtRiw2LVEiK0YnRi9GMkY1RjdGOUY7Rj1GP0ZgckZici1GUjYlUSNDMUYnRlVGV0Zcb0YvorLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY/LUkjbW9HRiQ2LVEiJ0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR1EmMC4wZW1GJy1GLDYtUSJ+RidGL0YyRjVGN0Y5RjtGPUY/L0ZCRkZGRC1GLDYtUSsmSW50ZWdyYWw7RidGL0YyRjUvRjhRJXRydWVGJ0Y5L0Y8Rk9GPUY/RkpGRC1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dGTy9GMFEnaXRhbGljRidGRy1GUjYlUSRzaW5GJy9GVkY0Ri8tSShtZmVuY2VkR0YkNiQtRiM2JEZRRi9GL0ZHLUZSNiVRJGNvc0YnRmZuRi9GZ25GRy1GLDYtUTAmRGlmZmVyZW50aWFsRDtGJ0YvL0YzUSZ1bnNldEYnL0Y2RmNvL0Y4RmNvL0Y6RmNvL0Y8RmNvL0Y+RmNvL0ZARmNvRkpGREZRRistRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZecEZHRkctSSZtZnJhY0dGJDYoLUYjNigtRiw2LVEqJnVtaW51czA7RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjIyMjIyMjJlbUYnL0ZFRmlwRlFGRy1JJW1zdXBHRiQ2JUZcby1GIzYkLUkjbW5HRiQ2JFEiMkYnRi9GLy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGZ25GL0ZecS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXHIvJSliZXZlbGxlZEdGNC1GLDYtUSIrRidGL0YyRjVGN0Y5RjtGPUY/RmhwRmpwRkstRmFwNigtRiM2JUZbcUZnbkYvRl5xRmdxRmpxRl1yRl9yRkdGX29GUUZHRmFyLUZSNiVRI0MyRidGVUZXLyUrZXhlY3V0YWJsZUdGNEYvThe second integral antiderivative somes out the same as 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apart from an additive 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The plot shows why the integral comes out negative.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv