121616949-math.209 - t = 0 and t = 2 ⇒ 6 The observation...

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9.5 Work 195 this: Z 3 1 v ( t ) dt = Z 3 1 16 t 2 + 5 dt = 446 3 . So now we see that another interpretation of the calculation 1 2 Z 3 1 16 t 2 + 5 dt = 1 2 446 3 = 223 3 is: total distance traveled divided by the time in transit, namely, average speed. It may also be helpful to interpret the average geometrically. The area under the velocity curve is Z 3 1 16 t 2 + 5 dt = 446 3 . The area under y = 223 / 3 over the same interval [1 , 3] is simply the area of a rectangle that is 2 by 223 / 3 with area 446 / 3. So the average height of a function is the height of the horizontal line that produces the same area over the given interval. Exercises 1. Find the average height of cos x over the intervals [0 , π/ 2], [ - π/ 2 , π/ 2], and [0 , 2 π ]. 2. Find the average height of x 2 over the interval [ - 2 , 2]. 3. Find the average height of 1 /x 2 over the interval [1 , A ]. 4. Find the average height of 1 - x 2 over the interval [ - 1 , 1]. 5. An object moves with velocity v ( t ) = - t 2 + 1 feet per second between t = 0 and t = 2. Find the average velocity and the average speed of the object between
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Unformatted text preview: t = 0 and t = 2. ⇒ 6. The observation deck on the 102nd floor of the Empire State Building is 1,224 feet above the ground. If a steel ball is dropped from the observation deck its velocity at time t is approximately v ( t ) =-32 t feet per second. Find the average speed between the time it is dropped and the time it hits the ground, and find its speed when it hits the ground. ⇒ 9.5 Work A fundamental concept in classical physics is work : If an object is moved in a straight line against a force F for a distance s the work done is W = Fs . EXAMPLE 9.12 How much work is done in lifting a 10 pound weight vertically a distance of 5 feet? The force due to gravity on a 10 pound weight is 10 pounds at the surface of the earth, and it does not change appreciably over 5 feet. The work done is W = 10 · 5 = 50 foot-pounds....
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