121616949-math.236 - integral 6 Set up the integral to...

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222 Chapter 9 Applications of Integration EXAMPLE 9.37 Let f ( x ) = p r 2 - x 2 , the upper half circle of radius r . The length of this curve is half the circumference, namely πr . Let’s compute this with the arc length formula. The derivative f is - x/ p r 2 - x 2 so the integral is Z r - r r 1 + x 2 r 2 - x 2 dx = Z r - r r r 2 r 2 - x 2 dx = r Z r - r r 1 r 2 - x 2 dx. Using a trigonometric substitution, we find the antiderivative, namely arcsin( x/r ). Notice that the integral is improper at both endpoints, as the function p 1 / ( r 2 - x 2 ) is undefined when x = ± r . So we need to compute lim D →- r + Z 0 D r 1 r 2 - x 2 dx + lim D r - Z D 0 r 1 r 2 - x 2 dx. This is not difficult, and has value π , so the original integral, with the extra r in front, has value πr as expected. Exercises 1. Find the arc length of f ( x ) = x 3 / 2 on [0 , 2]. 2. Find the arc length of f ( x ) = x 2 / 8 - ln x on [1 , 2]. 3. Find the arc length of f ( x ) = (1 / 3)( x 2 + 2) 3 / 2 on the interval [0 , a ]. 4. Find the arc length of f ( x ) = ln(sin x ) on the interval [ π/ 4 , π/ 3]. 5. Set up the integral to find the arc length of sin x on the interval [0 , π ]; do not evaluate the
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Unformatted text preview: integral. 6. Set up the integral to find the arc length of y = xe-x on the interval [2 , 3]; do not evaluate the integral. 7. Find the arc length of y = e x on the interval [0 , 1]. (This can be done exactly; it is a bit tricky and a bit long.) 9.10 Surface Area Another geometric question that arises naturally is: “What is the surface area of a vol-ume?” For example, what is the surface area of a sphere? More advanced techniques are required to approach this question in general, but we can compute the areas of some volumes generated by revolution. As usual, the question is: how might we approximate the surface area? For a surface obtained by rotating a curve around an axis, we can take a polygonal approximation to the curve, as in the last section, and rotate it around the same axis. This gives a surface...
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