MATA37H3 Lecture 12: Trigonometric Integrals

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cerisegiraffe538

12 Feb 2016

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Section 5. 4: warning: if we have an improper rational function (i. e. a rational function with deg(numerator) Deg(denominator)) then we must perform a polynomial long devision, plus algebra, before we apply a p. f. d. or other technique: let m, n, l, k n {0}. Consider the following integrals: sinn(x) cosm(x)dx, or sinn(x) cosm(x)dx tanl(x) seck(x)dx, or tanl(x) seck(x)dx (cid:90) b (cid:90) b a a: these are trigonometric intrgrals. We already know how to solve these: examples of trig. integrals: (cid:90) (cid:90) 0 (cid:90) (cid:90) (cid:90) (cid:90) sin5(x) cos2(x)dx sec3(x)dx (l = 0, k = 3) (cid:90) sin2(t) cot(t)dt = sin(t) cos(t)dt sin2(t) cos(t) sin(t) dt tan. 2 (x) sec(x)dx (this is not a trig. integral; note 1. 2 is not in n: example 1: cos2(x) sin5(x)dx (odd power) cos2(x) sin4(x) sin(x)dx cos2(x)(sin2(x))2 sin(x)dx cos2(x)(1 cos2(x))2 sin(x)dx. Let u = cos(x), du = sin(x)dx. Du = sin(x)dx (u2 2u4 + u6)du (cid:90) u7. 2 (cid:90) (cid:90) (cid:90) (cid:90) 1 cos(12x) (cid:90) sin2(6x)dx.

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Hello, I am having an extremely diffucult time understanding trig subs in Calc2. Can someone please fully expalin to me why the limits of integration change from 3sqrt3/2 to pi/3?....and than change once again from 1 to 1/2. I know you can plug the number somewhere to see if they change, but I'm not sure where to do this...or even WHY to do this? I still don't understand why we change limits of integration in other calc problems like u-sub. Can you please explain it like you're explaining it to someone who has no math skills. I really appreiciate a clear explanation without too much math terminology.

God bless and thank you!

1. (45 marks) Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answer. Unless otherwise stated, it is not necessary to simplify your answers in this question. a) Find the equation of the tangent plane to the graph of the function z = f(x,y) = sell at (0,1). Answer b) Evaluate (T, 0) if f(x,y) = sin(ry). Answer c) Graph of a function z = f(x,y) is given by: Which of the following level curves correspondes to this function. Answer d) If f(x,y) = e(3x – 2y), find lim - Math)-f(x,y) e) Find % if f(x, y) = x2 - y3 and 2 = r sin(t), y = r cos(t). Answer f) Evaluate the directional derivative of f(x, y) = !at (1, -1) in the direction of u =(2, 1). Answer 8) Does Simono converge? h) Sketch the direction field of the differential equation = 1-2y. i) Consider the differential equation ay = 15 - 31, y(0) = 0, and use Euler's method with step size 0.1 to estimate y(0.1). Answer j) Knowing that (-3,-3) is a critical point of f(x,y) = 2:3 + y2 – 4.cy -3.0, use the second derivative test to find out whether it is a local minimum, local maximum, or a saddle point. Answer k) Solve the initial value problem, dy dt 642 2y + cos(u) y(1) = . Answer 1) Is ū=(2,2, -1) perpendicular to v = (5,-4, 2)? Answer m) Find the area between the graph of y = 1 + sin(x) and y=1-sin(x) from 0 to 7. Answer n) Determine the lentgh of the curve y = 8/2,0 SIS 3. Evaluate and simplify your answer. Answer o) Express the volume of the solid obtained by rotating the bounded region that lies between curves y = (x-2)2 and y=1,150 $4 about the vertical line 2 = 5. Do not evaluate this integral. Answer

MATA37H3 Lecture 12: Trigonometric Integrals

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