Dr. Z’s Math251 Handout #16.2 (2nd ed.) [Line Integrals]By Doron ZeilbergerProblem Type 16.2a: Evaluate the line integral,ZCf(x, y) ds ,where C is some curve that the problem gives you in parametric form, or you have to representyourself (typically circles, line-segments, semicircles etc.).Example Problem 16.2a: Evaluate the line integral,ZCx2y ds ,where C is top half of the circle x2+ y2= 9.Steps Example1. Find the parametric equation of thecurve (x(t), y(t)), a ≤ t ≤ b, unless it isgiven by the problem.1. The parametric equation of a circle ofthe form x2+ y2= r2isx = r cos t , y = r sin t .So in our case we have r = 3 andx = 3 cos t , y = 3 sin t .Since it is the top half, t goes from 0 to π,so 0 ≤ t ≤ π.2. Computepx0(t)2+ y0(t)2.2. x0(t) = −3 sin t, y0(t) = 3 cos t, sopx0(t)2+ y0(t)2=p(−3 sin t)2+ (3 cos t)2=p9 sin2t + 9 cos2t =q9(sin2t + cos2t) =√9 = 3 .13. The line integral isZbaf(x(t), y(t))px0(t)2+ y0(t)2dt .Convert everything to the t-language andevaluate the t-integral from t = a to t = b.3.ZCx2y ds =Zπ0(3 cos t)2(3 sin t) · 3 dt81Zπ0cos2t sin t dt = 81−cos3t3π0= (−27)(cos3π − cos30) = 54 .Ans.: 54.Problem Type 16.2b: Evaluate the line integralZCP (x, y, z) dx + Q(x, y, z)dy + R(x, y, z) dz ,where C : x = x(t) , y = y(t) , z = z(t) , a ≤ t ≤ b.Example Problem 16.2b: Evaluate the line integralZCy dx + x dy + x2y√z dz ,where C : x = t3, y = t , z = t2, 0 ≤ t ≤ 1.Steps Example1. Get a (single variable) definite integral,in t, from t = a to t = b, by changingx, y, z to their expressions in terms of tand dx, dy, dz to x0(t)dt, y0(t)dt, z0(t)dt,respectively,ZCP (x, y, z) dx+Q(x, y, z)dy+R(x, y, z) dz .=Zba[P (x(t), y(t), z(t))x0(t) +Q(x(t), y(t), z(t))y0(t) +R(x(t), y(t), z(t))z0(t)] dt .1.ZCy dx + x dy + x2y√z dz=Z10t(3t2)dt + t3dt + (t3)2t√t2(2t)dt=Z10[4t3+ 2t9]dt .22. Evaluate the t-integration. 2.= t4+t10510== 1 +15− 0 =65.Ans.:65.Problem Type 16.2c: Evaluate the line integralZCF · dr ,where C is given by the vector function r(t).F(x, y, z) = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k ,r(t) = x(t)i + y(t)j + z(t)k , a ≤ t ≤ b .Example Problem 16.2c: Evaluate the line integralZCF · dr ,where C is given by the vector function r(t).F(x, y, z) = yzi + xzj + xyk ,r(t) = ti + t2j + t3k , 0 ≤ t ≤ 2 .Steps Example1. The desired line-integral equalsZCP dx + Q dy + R dz .Set-it up.1. Our integral isZCyz dx + xz dy + xy dz ,where x = t, y = t2, z = t3, 0 ≤ t ≤ 2.32. Evaluate this line integral like we didabove (16.2b).2.=Z20(t2)(t3) dt+(t)(t3)(2t) dt+(t)(t2)(3t2) dt ,=Z20[t5+ 2t5+ 3t5] dt=Z206t5dt = t620= 26− 06= 64 .Ans.: 64.A Problem from a previous FinalLet C be the line segment from (0, 1) to (3, 5), findRC2xy ds .Ans.: 55.Another Problem from a Previous Final(a) (4 points) Compute the surface integralZ ZS8 dS ,where S is the sphere (x −1)2+ (y + 4)2+ (z − 9)2= 100.(b) (4 points) Compute the triple integralZ Z ZE30 dV ,where E is the ball {(x, y, z) |(x − 1)2+ (y + 4)2+ (z − 9)2≤ 100 }.(c) (4 points) Compute the line integralZC3 ds ,where C is the circumference of the region {(x, y) |x2+ y2≤ 4 , y ≥ 0}.Ans.: 3200π, 40000π, 6π +