Exam 8: Using the Definite Integral

A bridge worker needs to pull a 10-meter uniform cable with mass 3 kg/meter up to the work platform.A second worker needs to pull an identical cable (which is one-half as long)up one-half the original distance.Will the second worker do one-half the work of the first worker? Assume that each cable is hanging straight down from its platform.

Focus on Engineering It's time for the School of Engineering class picture and you are the photographer! You stand at the origin with your camera and your classmates are strung out along the curve $y=e^{-x}$ from (0, 1)to $\left(5, e^{-5}\right)$ .You focus your camera at the average value of the distance to your classmates on the curve.Who is more in focus, the person at (0, 1)or the person at $\left(5, e^{-5}\right)$ ?

In a hydrogen atom in the unexcited state, the probability of finding the sole electron within x meters of the nucleus is given by $F(x)=\frac{4}{\left(a_{0}\right)^{3}} \int_{0}^{x} r^{2} e^{\frac{-2 r}{a_{0}}} d r$ , for $x \geq 0$ , where $a_{0}=5.29 \times 10^{-11}$ meters.What is its corresponding probability density function f(x)?

A professor gives the same 100-point final exam year after year and discovers that this students' scores tend to follow the triangular probability density function f(x)pictured below: Do the median and the mean both describe the same point on this probability density function? (All persons, places, and events in this story are fictitious.Any similarity to real persons or situations is purely coincidental.)