Consider the surface
(a) If
to evaluate the integral replace
(b) If
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- A parametric representation of the curve of intersection of the two surfaces x2 + 5y2 - z = 0 and z - 4y2 = 36 is given by the vector equation : r (t) = 6cosh(t) i + 2sinh(t) j + (36 - 16 sin²(t) ) K ,0sts 2n r (t) = 6cos(t) i + 2sin(t) j + (36 - 16 sin2(t) ) k , 0 st s 2n = cos(t) i + sin(t) j' + (36 - 16 sin²(t) ) k ,0 st s 2n %3D r (t) = 6cos(t) i + 6sin(t) j + 36(1 + 4 sin2(t) ) k ,O st s 2n r (t) = cos(t) i + 3sin(t) j + 36cos2(t) k ,0 st s 2Aarrow_forward(d) Find a vector function that represents the curve of intersection of the two surfaces: the xy cylinder x² + y² = 4 and the surface z = 4arrow_forwardConsider the surface given by the parametric vector function (image) 1. The graph of r (u, v) is the same graph of the surface z2 = x2 − y2 2. The surface is smooth in all its points. which is correct, incorrect or botharrow_forward
- a) A three dimensional motion of an object is given by the vector function r(t) = 4 cos t i+ 4 sin tj+5 k. Sketch the motion of the object when 0arrow_forwardConsider the function f(x, y) = (eª − x) cos(y). Suppose S is the surface z = f(x, y). (a) Find a vector which is perpendicular to the level curve of f through the point (4, 2) in the direction in which f decreases most rapidly. vector = (b) Suppose v = 77 +73+ ak is a vector in 3-space which is tangent to the surface S at the point P lying on the surface above (4, 2). What is a? a =arrow_forwardConsider the function f(x, y) = (eª — x) cos(y). Suppose S is the surface z = f(x, y). (a) Find a vector which is perpendicular to the level curve of f through the point (4, 2) in the direction in which f decreases most rapidly. vector = v (b) Suppose = 77 +73+ ak is a vector in 3-space which is tangent to the surface S at the point P lying on the surface above (4, 2). What is a? a = 175.9arrow_forward3. Find a vector function that represents the curve of intersection of the paraboloid z = r+y and the cylinder r+y = 16.arrow_forwardA parametric representation of the curve of intersection of the two surfaces x2 + 5y2 - z = 0 and 4y2 + z = 36 is given by the vector equation : r (t) = 6cos(t) i + 2sin(t) j + (36 - 16 sin?(t) ) k , O < t < 2n r (t) = cos(t) i + sin(t) j + (36 - 16 sin2(t) ) K , 0 st s 2n r (t) = 6cosh(t) + 2sinh(t) j + (36 - 16 sin2(t) ) k , 0 st< 2n r (t) = cos(t) i + 3sin(t) + 36cos?(t) k ,0 sts 2n r (t) = 6cos(t) i + 6sin(t) j + 36(1 - 4 sin?(t) ) K , 0 sts 27arrow_forwardFind both parametric and rectangular representations for the plane tangent to r(u,v)=u2i+ucos(v)j+usin(v)kr(u,v)=u2i+ucos(v)j+usin(v)k at the point P(4,−2,0)P(4,−2,0).One possible parametric representation has the form⟨4−4u⟨4−4u , , 4v⟩4v⟩(Note that parametric representations are not unique. If your first and third components look different than the ones presented here, you will need to adjust your parameters so that they do match, and then the other components should match the ones expected here as well.)The equation for this plane in rectangular coordinates has the form x+x+ y+y+ z+z+ =0arrow_forwardarrow_back_iosarrow_forward_ios
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage