Moments of Inertia In Exercises 53- 56, find
(a)
(b)
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Chapter 14 Solutions
Calculus (MindTap Course List)
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- Using Green's Theorem, find the outward flux of F across the closed curve C.F = (-5x + 2y) i + (6x - 9y) j; C is the region bounded above by y = -5x 2 + 250 and below by y=5x2 in the first quadrantarrow_forwardA region R consists of a square bounded by the lines x = -8, x = 8, y = 0, and y = -16 and a half disk bounded by the semicircle y = V 64 – x² and the line y = 0. Find the center of gravity, (x, y), of R. X = | 0 y Submit Answerarrow_forwardConsider the lamina in the shape of the triangle whose vertices are (0, 0), (L, 0), and (0,L) for some L ∈ R+. Calculate the moments of inertia, Ix and Iy, for this lamina whose density is proportional to the square of the distance from the vertex opposite the hypotenuse.arrow_forward
- Let F = -9zi+ (xe#z– 2xe**)}+ 12 k. Find f, F·dĀ, and let S be the portion of the plane 2x + 3z = 6 that lies in the first octant such that 0 < y< 4 (see figure to the right), oriented upward. Z Explain why the formula F · A cannot be used to find the flux of F through the surface S. Please be specific and use a complete sentence.arrow_forwardSketch the region enclosed by y = 0, y = (x + 1)3, and y = (1 − x)3, and find its centroid.arrow_forwardWhere is the centroid located?arrow_forward
- Exercise 3 TRIPLE INTEGRAL IN CYLINDRICAL COORDINATES. Consider the region W that lies between the sphere x2 +y? + z? = 4, above the plane z = 0, and inside the cylinder a2 + y? = 1. (i) Sketch the region W. (ii) Use cylindrical coordinates to integrate f(r, y, z) = z over W.arrow_forwardCalculate the curl(F) and then apply Stokes' Theorem to compute the flux of curl(F) through the surface of part of the cone √x² + y2 that lies between the two planes z = 1 and z = 8 with an upward-pointing unit normal, vector using a line integral. F = (yz, -xz, z³) (Use symbolic notation and fractions where needed.) curl(F) = flux of curl(F) = [arrow_forwardDetermine the x- and y-coordinates of the centroid of the shaded area. 7" y -8" Answers: (X,Y)= (i 3" 13" 8" " x ) in.arrow_forward
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