INTEGRAL CALCULU Math

Math 252 – Homework #5 Name ___________________ Due: Wednesday, 31 July 2019

1) Determine the equation of the curve, through the point (1,1) whose length from x = 1 to x = 3 is given by

L = 1 + 1

x 2 dx

1

2

ò

2) Suppose a 24-meter length of wire has density function 𝑓(𝑥) = ( 𝑥

6 ) 1/2

, measured in grams per meter

a) What is the total mass of the wire?

b) Where would we have to cut the wire to get a piece that weighs 4 grams?

3) Consider Hooke's Law: The force required to keep a spring in a compressed or stretched position x units from the spring’s equilibrium position is F(x) = kx.

Calculate the work required, in joules, to stretch a spring 0.6 meters beyond its equilibrium position for each of the following scenarios.

a) The spring requires 30 Newtons of force to hold it 0.1 m from its equilibrium position.

b) The spring requires 6 Joules of work to stretch the spring 0.1 meter from its equilibrium position.

c) Explain how the above two problems are similar and how they are different and why you have to approach them differently.

4) A conical tank has height 8 meters and radius 6 meters, oriented as shown below. The tank

contains 7 meters of water. Set up an integral to find the work done in pumping all the water out the spout of which is 2 meters above the top of the tank.