Question

Can someone please help me solve these problems without skipping steps I need to understand the process. Thank you

If V xy 2xy2 + yz, evaluate V dr between A (0, 0, 0) and B (2, 1, -3) along the curve with parametric equations x 2t, y t, z 3t3 If F-ry® i + yz2 j + zx2 k, evaluate | F-dr along the curve x 3u2, y u, z 2u between A (3,-1, 2) and B (3, 1,2) Evaluate FdV where F 3i+z+2yk and V is the region bounded by the planes z 0, z 3 and the surface x2 +y 4.

Answer 1

V = x^3 y + 2x y^2 + yz We have to find integral V dr between A(0, 0, 0) and B(2, 1, -3) along curve with parametric equations x = 2t , y = t^2, z = -3t^3. Therefore dx = 2 dt, dy = 2t dt, dz = -9t^2 dt. Also range of t is between 0 and 1. Since the range of x is between 0 and 2. and that of y and z are between 0 and 1 and Therefore integral_c V dr = integral^1_c (x^3 y + 2x y^2 + yz)(dx i + dy j + dz k) = integral^1_0 (8t^3 . t^2 + 2 . 2t . t^4 + t^2(-3t^3))(2dt i + 2t dt j -9t^2 dt z) = integral^1_0 9t^5(2dt i + 2t dt j -9t^2 dt k) = [18t^6/6 i + 18t^7/7 j -81t^8/8 k]^1_0 = 3 i + 18/7 j -81/8 k. Here F = x^2 y^3 i + y z^2 j + z x^2 k.