Download Math 20B. Lecture Examples. Section 7.3. Trigonometric integrals†

(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.