**Unformatted text preview: **x 2 + y 2 = 4 and the circle x 2 + y 2 = 1, above the x-axis. Find the centroid. ⇒ 12. A thin plate lies in the region between the circle x 2 + y 2 = 4 and the circle x 2 + y 2 = 1 in the ﬁrst quadrant. Find the centroid. ⇒ 9.7 Kinetic energy; improper integrals Recall example 9.14 in which we computed the work required to lift an object from the surface of the earth to some large distance D away. Since F = k/x 2 we computed Z D r k x 2 dx =-k D + k r . We noticed that as D increases, k/D decreases to zero so that the amount of work increases to k/r . More precisely, lim D →∞ Z D r k x 2 dx = lim D →∞-k D + k r = k r . We might reasonably describe this calculuation as computing the amount of work required to lift the object “to inﬁnity,” and abbreviate the limit as lim D →∞ Z D r k x 2 dx = Z ∞ r k x 2 dx....

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- Fall '07
- JonathanRogawski
- Math, Calculus, Improper Integrals, Integrals, Energy, lim, 4 meters, 10 meters