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(7/1/09) Math 20B. Lecture Examples. Section 7.3. Trigonometric integrals† Rule 1 (a) To find the integral of y = sinn x cosm x where n is an odd positive integer and m is any constant, use the Pythagorean identity sin2 x + cos2 x = 1 to change all but one of the sines to cosines. Then use the substitution u = sin x, du = cos .x dx. (b) If m is an odd positive integer and n is any constant, use sin2 x + cos2 x = 1 to change all but one of the cosines to sines and use the substitution u = cos x, du = −sin x dx. Example 1 Perform the integration Answer: Example 2 Z sin2 x cos3 x dx. 1 sin2 x cos3 x dx = 3 sin3 x − 1 sin5 x + C 5 Evaluate Answer: Rule 2 Z Z Z π/2 √ sin3 x cos x dx. 0 π/2 √ 8 sin3 x cos x dx = 21 0 To find the integral of y = sinn cosm x where n and m are both nonnegative, even integers, use the double angle formulas sin2 θ = 1 2 [1 − cos(2θ)] and cos2 θ = 12 [1 + cos(2θ)] one or more times to express the integrand as a linear combination of functions y = cos(kx) with positive integers k. Example 3 Find the volume of the solid that is generated when the region between y = sin x and the x-axis for 0 ≤ x ≤ π is rotated about the x-axis. π2 Answer: Figures A3a and A3b • [Volume] = 1 2 y y = sin x 1 sin x x π x −1 The rotated region Figure A3a † Lecture The cross section at x Figure A3b notes to accompany Section 7.3 of Calculus, Early Transcendentals by Rogawski. 1 Math 20B. Lecture Examples. (7/1/09) Section 7.3, p. 2 Rule 3 (Integrals of products of sin (ax) and cos (bx)) Integrals of y = cos(ax)cos(bx), y = cos(ax)sin(bx), and y = sin(ax)sin(bx) with unequal constants a and b are most easily evaluated by using the product identities, sin θ sin ψ= 12 [cos(θ − ψ) − cos(θ + ψ)] (1) cos θ cos ψ= 12 [cos(θ − ψ) + cos(θ + ψ)] (2) sin θcosψ= 12 [sin (θ − ψ) + sin(θ + ψ)]. (3) If necessary, the results of applying Rule 3 can be expressed in terms of sin(ax) and cos(bx) by using the sum and difference identities, Example 3 cos(θ ± ψ) = cos θ cos ψ ∓ sin θ sin ψ (4) sin(θ ± ψ) = sin θ cos ψ ± cos θ sin ψ. (5) Perform the integration Answer: Z Z sin(5x) cos(2x) dx. 1 1 sin(5x) cos(2x) dx = − 6 cos(3x) − 14 cos(7x) + C Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡ Section 8.2: Examples 1–3 ‡ The 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.