121616949-math.219

# 121616949-math.219 - 9.7 Kinetic energy improper integrals...

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9.7 Kinetic energy; improper integrals 205 Exercises 1. A beam 10 meters long has density σ ( x ) = x 2 at distance x from the left end of the beam. Find the center of mass ¯ x . 2. A beam 10 meters long has density σ ( x ) = sin( πx/ 10) at distance x from the left end of the beam. Find the center of mass ¯ x . 3. A beam 4 meters long has density σ ( x ) = x 3 at distance x from the left end of the beam. Find the center of mass ¯ x . 4. Verify that Z 2 x arccos x dx = x 2 arccos x - x 1 - x 2 2 + arcsin x 2 + C . 5. A thin plate lies in the region between y = x 2 and the x -axis between x = 1 and x = 2. Find the centroid. 6. A thin plate fills the upper half of the unit circle x 2 + y 2 = 1. Find the centroid. 7. A thin plate lies in the region contained by y = x and y = x 2 . Find the centroid. 8. A thin plate lies in the region contained by y = 4 - x 2 and the x -axis. Find the centroid. 9. A thin plate lies in the region contained by y = x 1 / 3 and the x -axis between x = 0 and x = 1. Find the centroid. 10. A thin plate lies in the region contained by x + y = 1 and the axes in the first quadrant. Find the centroid.
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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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