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MATH 1310.03F A Exam 1 Thursday, October, 12 2000

NAME:

Student Number:tex2html_wrap_inline219 No.tex2html_wrap_inline221Marks tex2html_wrap_inline223


Instructions:

1. You have 50 minutes for this exam. You are not allowed to use any calculators.

2. This exam contains 3 questions and has a total of 100 marks.

3. Show all of your work. Your work must justify the answer that you give.

tex2html_wrap_inline231(20 Marks)Find the volume of the solid generated by revolving the region bounded by the parabolas tex2html_wrap_inline233 and tex2html_wrap_inline235 about the x-axis.

Solution: Let tex2html_wrap_inline239. Then x=0 and x=1. We draw the following graph.

(Disc Method)

Region: tex2html_wrap_inline233 and tex2html_wrap_inline249 for tex2html_wrap_inline251.

Axis of revolution: x-axis.

The volumetex2html_wrap_inline255.

(Shell Method ) Region: tex2html_wrap_inline259 and tex2html_wrap_inline261tex2html_wrap_inline263.

Axis of revolution: x-axis.

The volume tex2html_wrap_inline267.

tex2html_wrap_inline269(20 Marks)Evaluate tex2html_wrap_inline271.

Solution: Let tex2html_wrap_inline273. Then we havetex2html_wrap_inline275
displaymath277

Hence tex2html_wrap_inline279. Sincetex2html_wrap_inline277 and tex2html_wrap_inline279 is continuous, we have
displaymath281

tex2html_wrap_inline283(60 Marks)Evaluate each of the following indefinite integrals.

 (atex2html_wrap_inline287
(btex2html_wrap_inline291
(atex2html_wrap_inline287 (btex2html_wrap_inline291
 (ctex2html_wrap_inline303
(dtex2html_wrap_inline307
(ctex2html_wrap_inline311 (dtex2html_wrap_inline307

Solution: (a) Let tex2html_wrap_inline319. Then tex2html_wrap_inline321 andtex2html_wrap_inline323. This implies
displaymath325
 

eqnarray126

(b)
eqnarray155

(ctex2html_wrap_inline331

(d)
eqnarray177

This impliestex2html_wrap_inline333 and
displaymath335




Kunquan Lan

Thu Oct 12 12:53:31 EDT 2000