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Approximate the arc length of the graph of the function over the interval [0, 4] in four ways. (a) Use the Distance Formula to find the distance between the endpoints of the arc. (b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when x=0, x=1, x=2, x=3, and x=4, Find the sum of the four lengths. (c) Use Simpson’s Rule with n=10 to approximate the integral yielding the indicated arc length. (d) Use the integration capabilities of a graphing utility to approximate the integral yielding the indicated arc length. f(x) = x³
Moment of area is summation of area time's distance to an axis. It is a measure of the distribution of the area of a shape in relationship to an axis.
To summarize, for symmetric surfaces(along an axis) the moment of area along that axis() will be zero because the area on one side of the axis will be exactly equal in magnitude and opposite in direction to the area on the other. For surfaces with higher area on one side of the axis(assuming equidistant from axis), the center will shift towards the side with higher area because that side will contribute more moment than the side with less area.
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