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HH_Sec_6_2 - Math 10B Lecture Examples Section 6.2...

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Unformatted text preview: (9/30/08) Math 10B. Lecture Examples. Section 6.2. Constructing antiderivatives analytically † Example 1 (a) Find the antiderivative integraldisplay parenleftbigg 3 √ x + 4 x 2- 3 parenrightbigg dx . (b) Check the answer by differentiation. Answer: (a) integraldisplay parenleftbigg 3 √ x + 4 x 2- 3 parenrightbigg dx == 2 x 3 / 2- 4 x- 1- 3 x + C (b) d dx (2 x 3 / 2- 4 x- 1- 3 x ) = 3 √ x + 4 x 2- 3 Example 2 What is the value of the integral integraldisplay 1- 1 x 2 dx ? Answer: integraldisplay 1- 1 x 2 dx = 2 3 Example 3 Evaluate integraldisplay 2 1 ( 4x 1 / 3 + 6x- 2 ) dx . Answer: integraldisplay 2 1 (4 x 1 / 3 + 6 x- 2 ) dx = 3(2 4 / 3 ) Example 4 Find the area of the region bounded by the curve y = 3x 2- x 3 and the x-axis. Answer: Figure A3 • [Area] = 27 4 x 1 2 y 2 4 y = 3 x 2- x 3 Figure A3 Example 5 Suppose that the temperature in a room is 50 ◦ F at time t = (hours) and that the rate of change of the temperature is r = 12t 2- 4t 3 degrees per hour at time t for ≤...
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HH_Sec_6_2 - Math 10B Lecture Examples Section 6.2...

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