HH_Sec_7_3 - (9/30/08) Math 10B. Lecture Examples. Section...

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Unformatted text preview: (9/30/08) Math 10B. Lecture Examples. Section 7.3. Tables of integrals† Example 1 Find the area of the region between y = 1 and the x-axis for x(3x + 6) 1 ≤ x ≤ 3. Use the following formula from a table of integrals: 1 (ax + b)(cx + d) dx = 1 ax + b ln + C for a = 0, c = 0, ad − bc = 0. ad − bc cx + d 9 Answer: [Area] = 1 ln( 5 ) 6 Example 2 Find a formula for the function y = F(x) such that F (x) = x2 ex for all x and F(0) = 3. Use the following formula from a table of integrals: x2 eax dx = Answer: F (x) = (x2 − 2x + 2)ex + 1 2 2 12 x − 2 x+ 3 a a a eax + C Example 3 Find the area of the region between the x-axis and y = sin x cos x for 1 0 ≤ x ≤ 2 π .Use the following formula from a table of integrals: sin(ax) cos(ax) dx = Answer: [Area] = 1 2 1 sin2 (ax) + C 2a Example 4 Find the antiderivative table of integrals: ex dx. Use the following formula from a 4 − (ex )2 u+a 1 1 ln +C du = 2a u−a a2 − u2 ex Answer: 4 − (ex )2 dx = 1 4 ln ex + 2 +C ex − 2 Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡ Section 8.5: Examples 1 through 3 notes to accompany Section 7.3 of Calculus by Hughes-Hallett et al. chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course. ‡ The † Lecture 1 ...
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