MAT1505-03/04 15F Test 1 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 when appropriate). Indicate where technology is used and what type (Maple, GC).
Optional parts are only for students who have completed the remaining questions to their satisfaction.1. 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 down a definite integral for the volume of the solid of revolution which results from revolving the region LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= around the axis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRIjFGJ0Y5Rjk=. Support your choice of integrand and limits of integration with a fully labeled diagram shading the region of integration and including an appropriately labeled typical cross-section and identifying the ingredients in your integrand in the diagram. Then evaluate your integral with technology.2. The temperature in degrees Fahrenheit for a 12 hour period from 9am to 9pm is 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, with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= the time in hours starting at 9am. a) Find the average temperature during this period exactly and to the nearest tenth of a degree, showing all steps in the hand integration process.
b) Find the times LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= when the temperature equals the average temperature.c) Optional. Find the clock times (as in 10:31am) to the nearest minute when the temperature equals the average temperature.3. Given the velocity profile 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUZLNiVRIlJGJ0ZORlFGLw== , write down a definite integral for its average value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EpYHZfX2F2Z2BGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= and rewrite this integral in terms of the dimensionless variable 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 and simplify to the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJVRidGL0YyRjk= where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a dimensionless integral involving no parameters. What is the exact and numerical value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ? [Use Maple to evaluate this final integral.]b) Optional: What is the value of the ratio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiNRKWB2X19hdmdgRictRiM2JS1GLzYjUSlgdl9fbWF4YEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGQi8lKWJldmVsbGVkR1EmZmFsc2VGJy9GO1Enbm9ybWFsRic= (exact and to 4 decimal places), where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EpYHZfX21heGBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= is the maximum value of the velocity on this interval? Use this result to make a simple sketch of the velocity profile and its average value (as a horizontal line).4. What does the Fundamental theorem of Calculus say about evaluating this limit: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Jy1JI21pR0YkNiVRImhGJy8lJ2l0YWxpY0dGQi9GM1EnaXRhbGljRictRi82LVEtJnJpZ2h0YXJyb3c7RidGMi9GNlEmZmFsc2VGJy9GOUZZL0Y7RkIvRj1GWS9GP0ZZL0ZBRlkvRkRGWS9GRlEsMC4yNzc3Nzc4ZW1GJy9GSUZbby1JI21uR0YkNiRRIjBGJ0YyLyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjIvJSxhY2NlbnR1bmRlckdGWS1JJm1mcmFjR0YkNigtRl5vNiRRIjFGJ0YyLUYjNiVGTUZRRlMvJS5saW5ldGhpY2tuZXNzR0ZbcC8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZicC8lKWJldmVsbGVkR0ZZLUkobXN1YnN1cEdGJDYnLUYvNi1RKyZJbnRlZ3JhbDtGJ0YyRlhGWkZlbkZmbi9GP0ZCRmhuRmluRkUvRklGRy1GIzYoLUZebzYkUSIyRidGMkZRLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGQkZhb0ZTLUYjNipGYXEtRi82LVEiK0YnRjJGWEZaL0Y7RllGZm5GZ25GaG5GaW4vRkZRLDAuMjIyMjIyMmVtRicvRklGYHJGTUZRRmRxRmdxRmFvRlMvJTFzdXBlcnNjcmlwdHNoaWZ0R0Zgby8lL3N1YnNjcmlwdHNoaWZ0R0Zgby1JJm1zcXJ0R0YkNiMtRiM2J0Zpb0Zbci1JJW1zdXBHRiQ2JS1GTjYlUSJ4RidGUUZTLUYjNiYtRl5vNiRRIjNGJ0YyRlFGYW9GU0ZickZhb0YyLUZONiNRIUYnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjVlbUYnLyUmZGVwdGhHRl50LyUqbGluZWJyZWFrR1ElYXV0b0YnLUYvNi1RMCZEaWZmZXJlbnRpYWxEO0YnRjJGNUY4RjpGPEY+L0ZBRjdGQ0ZFRl5xRl5zRmFvRjI=. Explain how you evaluated it.
[Hint: let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be an antiderivative of this integrand.]solutionunused1b) Optional: Write the equation of the straight line which connects the two points of intersection and which cuts this region in two parts. Write down a fully simplified definite integral for the area of the upper part. Evaluate it with technology.a) 2. Consider the definite integral: 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 .
Evaluate this integral step by step. Confirm that your result agrees with Maple.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYtLUY2NiQtRiM2Jy1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSgmbWludXM7RidGQi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSi8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZZLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR0ZKRkJGQi1GRTYtUSJ+RidGQkZIRktGTUZPRlFGU0ZVL0ZYUSYwLjBlbUYnL0ZlbkZfby1JJW1zdXBHRiQ2JS1GRTYtUS8mRXhwb25lbnRpYWxFO0YnRkIvRklRJnVuc2V0RicvRkxGaG8vRk5GaG8vRlBGaG8vRlJGaG8vRlRGaG8vRlZGaG9GXm8vRmVuUSwwLjExMTExMTFlbUYnLUYjNipGPkZERmZuRi8vJStmb3JlZ3JvdW5kR1ErWzAsMTYwLDgwXUYnLyUscGxhY2Vob2xkZXJHRjEvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjFGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRkU2LVEiLEYnRkJGSC9GTEYxRk1GT0ZRRlNGVUZeby9GZW5RLDAuMzMzMzMzM2VtRidGZm4tRkU2LVEiPUYnRkJGSEZLRk1GT0ZRRlNGVS9GWFEsMC4yNzc3Nzc4ZW1GJy9GZW5GZ3EtRj82JEZccUZCLUZFNi1RIy4uRidGQkZIRktGTUZPRlFGU0ZVRldGYG9GPkZpbkZCRkItRkU2LVEiO0YnRkJGSEZgcUZNRk9GUUZTRlVGXm9GaHFGW28tSShtc3Vic3VwR0YkNictRkU2LVErJkludGVncmFsO0YnRkJGZ29GaW8vRk5GMUZbcC9GUkYxRl1wRl5wRl5vRmBvLUYjNiRGaXFGQi1GIzYkLUYsNiVRImFGJ0YvRjJGQkZqcC8lL3N1YnNjcmlwdHNoaWZ0R0ZccS1GNjYkLUYjNiZGPkZELUkmbWZyYWNHRiQ2KEZmbi1GIzYlRl1zRi9GMi8lLmxpbmV0aGlja25lc3NHRkEvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGX3QvJSliZXZlbGxlZEdGSkZCRkJGW28tRiM2JS1GLDYjUSFGJy1GYm82JUZkby1GIzYqRj5GREZmc0YvRmNwRmZwRmhwRjJGanBGQkZbby1GRTYtUTAmRGlmZmVyZW50aWFsRDtGJ0ZCRmdvRmlvRmpvRltwRlxwRl1wRl5wRl5vRmBvRmZuRmluRkI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUYjNictSSNtbkdGJDYkUSIwRidGMi8lJ2l0YWxpY0dGPC8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUscGxhY2Vob2xkZXJHRjwvRjNRJ2l0YWxpY0YnLUYjNictRk02JFEiMUYnRjJGUEZSRlVGVy8lMXN1cGVyc2NyaXB0c2hpZnRHRk8vJS9zdWJzY3JpcHRzaGlmdEdGTy1JKG1mZW5jZWRHRiQ2JC1GIzYnRmVuLUYvNi1RKCZtaW51cztGJ0YyL0Y2USZmYWxzZUYnL0Y5RmVvL0Y7RmVvL0Y+RmVvL0ZARmVvL0ZCRmVvL0ZERmVvL0ZGUSwwLjIyMjIyMjJlbUYnL0ZJRl1wLUkjbWlHRiQ2JVEieEYnRlBGVy8lK2V4ZWN1dGFibGVHRmVvRjJGMi1GLzYtUSJ+RidGMkZkb0Zmb0Znb0Zob0Zpb0Zqb0ZbcEZFRkgtRmBwNiVRJHNpbkYnL0ZRRmVvRjItRl1vNiQtRiM2JUZfcEZjcEYyRjJGZXAtRi82LVEwJkRpZmZlcmVudGlhbEQ7RidGMkY1RjgvRjtGN0Y9L0ZARjdGQUZDRkVGSEZfcEZjcEYyLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYuLUY2NiQtRiM2Ji1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSgmbWludXM7RidGQi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSi8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZZLUYsNiVRInhGJ0YvRjJGQkZCLUZFNi1RIn5GJ0ZCRkhGS0ZNRk9GUUZTRlUvRlhRJjAuMGVtRicvRmVuRl1vLUYsNiVRJHNpbkYnL0YwRkpGQi1GNjYkLUYjNiRGZm5GQkZCLUZFNi1RIixGJ0ZCRkgvRkxGMUZNRk9GUUZTRlVGXG8vRmVuUSwwLjMzMzMzMzNlbUYnRmZuLUZFNi1RIj1GJ0ZCRkhGS0ZNRk9GUUZTRlUvRlhRLDAuMjc3Nzc3OGVtRicvRmVuRmFwLUY/NiRRIjBGJ0ZCLUZFNi1RIy4uRidGQkZIRktGTUZPRlFGU0ZVRldGXm9GPi8lK2V4ZWN1dGFibGVHRkpGQkZCRmlwRkI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkaW50RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNi0tRjY2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RKCZtaW51cztGJ0ZCLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZKLyUpc3RyZXRjaHlHRkovJSpzeW1tZXRyaWNHRkovJShsYXJnZW9wR0ZKLyUubW92YWJsZWxpbWl0c0dGSi8lJ2FjY2VudEdGSi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlktRiw2JVEieEYnRi9GMkZCRkItRkU2LVEifkYnRkJGSEZLRk1GT0ZRRlNGVS9GWFEmMC4wZW1GJy9GZW5GXW8tRiw2JVEkc2luRicvRjBGSkZCLUY2NiQtRiM2JEZmbkZCRkItRkU2LVEiLEYnRkJGSC9GTEYxRk1GT0ZRRlNGVUZcby9GZW5RLDAuMzMzMzMzM2VtRidGZm4tRkU2LVEiPUYnRkJGSEZLRk1GT0ZRRlNGVS9GWFEsMC4yNzc3Nzc4ZW1GJy9GZW5GYXAtRj82JFEiMEYnRkItRkU2LVEjLi5GJ0ZCRkhGS0ZNRk9GUUZTRlVGV0Zeb0Y+RkJGQi8lK2V4ZWN1dGFibGVHRkpGQg==JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRictRiw2JVEld2l0aEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEiOkYnRjlGOy9GP0Y9RkBGQkZERkZGSC9GS0ZPRk1GZW5GOQ==1. 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=line through endpoints not 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSQxNDRGJy9GM1Enbm9ybWFsRictRj42JFEiNUYnRkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRksvJSliZXZlbGxlZEdRJmZhbHNlRidGQQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=2. average 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1:15pm, 7:35pmLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=3. change of variable in definite integralThe average value of 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 over the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUZLNiVRIlJGJ0ZORlFGLw== isQykqKEkiQ0c2IiIiIkkiUkdGJSEiIi1JJGludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKiYsJiokRiciIiNGJiokSSJyR0YlRjJGKEYmLUkkZXhwR0YrNiMsJComRjRGMkYnISIjRihGJi9GNDsiIiFGJ0YmRiYqKEkiJUdGJUYmRiRGKEYnRjpGJi1JKXNpbXBsaWZ5R0YrNiNGP0YmLUkmZXZhbGZHRixGQg==This integral is equal to 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the original constant coefficient of the integrand. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0Yv is the dimensionless number definite integral, whose approximate value is given. Setting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= in the original integrand (without the multiplying constant) gives that dimensionless integral. Maple can change variables with the integration methods tutor, a bit clunky.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4. fundamental theorem of 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=pledgeWhen you have completed the exam, please read and sign the dr bob integrity pledge and hand this test sheet in on top of your answer sheets as a cover page, with the first test page facing up:"During this examination, all work has been my own. I give my word that I have not resorted to any ethically questionable means of improving my grade or anyone else's on this examination and that I have not discussed this exam with anyone other than my instructor, nor will I until after the exam period is terminated for all participants."
Signature: Date: