Cycloid Consider one arch of the cycloid
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Calculus
- Polar curves C2: r = 4cosθ and C3 : r = 4sinθ. Solve for the slope of the tangent line at the non-pole point on C2 which intersects C3 . Set up an integral giving the area of the region inside both C2 and C3 .arrow_forwardLength of curves Use a scalar line integral to find the length ofthe following curve. r(t) = ⟨30 sin t, 40 sin t, 50 cos t⟩ , for 0 ≤ t ≤ 2πarrow_forwardWalking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane. a.In each case, find z’(t). b.Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). z = 4x2 - y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2πarrow_forward
- Walking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane. a. In each case, find z'(t). b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). z = x2 + 4y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2πarrow_forwardFind the volumes of the solids The solid lies between planes perpendicular to the x-axis at x = π/4 and x = 5π/4. The cross-sections between these planes are circular disks whose diameters run from the curve y = 2 cos x to the curve y = 2 sin x.arrow_forwardArc length of the curve from point P to Q. x^2=(y-4)^3, P(1,5), Q(27,13)arrow_forward
- A lamina occupies the part of the disk x2 + y2 ≤ 49 in the first quadrant. Find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin. Hint: use polar coordinatesarrow_forwardOutward normal to a sphere Show that | tu x tv | = a2 sin u for a sphere of radius a defined parametrically by r(u, v) = ⟨a sin u cos v, a sin u sin v, a cos u⟩ , where 0 ≤ u ≤ π and 0 ≤ v ≤ 2π.arrow_forwardThe force F newtons acting on a body at distance x metres from a fixed point is given by F = 2x2 + 3x. Determine the work done when the body moves from x = 1 m to x = 3 m. (b) Calculate the electric charge that passes in the first 2 seconds in a circuit when a current of i = 2sin(p/2)t flows. (c) Determine the area bounded by the curve y = x(2- x) and the x-axis. Thanksarrow_forward
- Parametrize the sine curve y = sinx using the parametrization r(t)=(t,sint), t ∈ R. (a) show that this curve is smooth. (b) Compute the curvature, and find all points where the curvature is zero. What geometric property do all those points share? (c) Without doing any computations, explain why the torsion of this curve must be identically zero. (d) Rotating the plane 30◦ is an isometry, which transforms the original sine curve into the curve parametrized by r(t) = √3t −sint 2 , t + √3 sint 2 , t ∈ R. Compute the speed, curvature, and torsion of this curve, and compare them to those of the original curve. (e) By graphing the curve, decide if it is the graph of some function y = f(x). (f) In single-variable calculus, you studied the qualitative properties of the graphs of functions y = f(x). In particular, you characterized the maxima, minima, and inflection points in terms of the vanishing of certain derivatives of the function f(x). Using the earlier parts of this problem to supply…arrow_forwardArc length Find the arc length of the following curves on the given interval. x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2πarrow_forward(a) Find the exact area of the surface obtained by rotating the curve y = e^x about thex-axis over the interval 0 ≤ x ≤ 1.(b) Determine the length of the parametric curve given by the following set ofparametric equations.x = 3 cos t − cos 3t, y = 3 sin t − sin 3t, 0 ≤ t ≤ πYou may assume that the curve traces out exactly once for the given range of t.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage