Integrate the differential one-form ayr dy over the indicated closed curves (a) The circle x2 + y...
Question 2 (30 points) Integrate f(x, y,2) xzv2-z2 - y2 over the path C, which consists of two curves, C1 and C2 from (1, 0, 0) to (1,0, 0), then to (-1,3, 0). Curve C1 is only half of the circle2 Curve C2 is a straight-line segment. The parametric equation for G is G: r! (t)-cos t î + sin t k, 0 π Find the line integral: Jcf(x. y,z)ds - (25 points) C2 (-1,3,0)
Question 2 (30 points) Integrate...
Use Symbolics in Matlab to integrate the following ordinary differential equation. dy/dx + y cos(2x) = (1/2) sin 3x
(b) Let C be the closed curve formed by intersecting the cylinder x2 +y= 1 with the plane x z= 2. Let the tangent to the curve from above. point in the anti-clockwise direction when viewed Calculate the line integral (e (e sin y+ 4) dy+(e(cos z+ sin z)+ay) dz. cos x2yz) dx +
(b) Let C be the closed curve formed by intersecting the cylinder x2 +y= 1 with the plane x z= 2. Let the tangent to the...
(1) Evaluate the following line integrals in R3. r +yds for C the line segment from (0, 1,0) to (1, 0,0) for C the line segment from (0,1,1) to (1,0,1). for C the circle (0, a cos t, a sin t) for O (iv) 2π, with a a positive constant. t for C the curve (cost +tsint,sint tcost, 0) for Osts v3 (Hint for (i): use the parametrization (z, y, z) = (t, 1-t, 0) for 0 1) t
(1)...
3. 8p] Show that the force field F(x,y, z) sin y, x cos y + cos z, -y sin z) is conservative and use this fact to evaluate the work done by F in moving a particle with unit mass along the curve C with parametrization r(t (sin t, t, 2t), 0 <t<T/2. 4. 8p] A thin wire has the shape of a helix x = sin t, 0 < t < 27r. If the t, y = cos t,...
Triple Integration
Problems.
1. Integrate zdV JJ w where ll' is enclosed by the planes z = 0 and cylinders x2 + y2 4 and x2 + y,: 9 = x+9+ 3 and by the 2. Integrate where E is bounded by the zu-plane and the hemispheres z/9-2y2 and z = V/10-22-27 Change the order of integration and evaluate x3 sin(уз)dydx. 0 Jr2
1. Integrate zdV JJ w where ll' is enclosed by the planes z = 0 and cylinders...
(7.5 points) Let C be the oriented closed space curve traced out by the parametrization r(t) = (cost, sint, sin 2t), 0<t<27 and let v be the vector field in space defined by v(x, y, z) = (et - yº, ey + r), e) (a) Show that C lies on the cylinder x2 + y2 = 1 and the surface z = 2cy. (b) This implies that C can be seen as the boundary of the surface S which is...
So derive (by hand) the closed-form solutions to the following three problems using this approach: dy (a)| +5y= 6 cos(t) sin(t); y(0)=5 (b) | dy-y=cos2(t): y(0)=1 dy sin 0 (c)-+ 3y ;y(0) = 2 dt 2S1
Help Entering Answers (1 point) Verify that Stokes' Theorem is true for the vector field F = yzi - yj + xk and the surface S the part of the paraboloid z= 4 a2 ythat lies above the plane z = 3, oriented upwards. curl FdS To verify Stokes' Theorem we will compute the expression on each side. First compute S curl F = Σ <0,y-1,-z> curl F.dS Σ dy dπ (y-1)-2y)+z where 3 -sqrt(9-x^2) Σ 3 sqrt(9-x^2) curl F...
I do NOT need part a. I really need help on b,c,d,and e though!
Thank you
2. Evaluate the line integral where C is the given curve: BE SURE THAT YOU PARAMETERIZE EACH CURVE! (a) ez dr where C is the arc of the curve z = y3 from (-1,-1) to (1,1); (b) 2,2 d_T + y2 dy where C consists of the arc of the circle x2 + y2-4 from (2,0) to (0,2) followed by the line segment from...