Calculus 1
Fall 2013
Lab 12: Definite Integrals – Part 1
Work with your group on the context assigned to you. We encourage you to collaborate both in and
out of class, but you must write up your responses individually.
Context 1: For a constant force* F to move an object a distance d requires an amount of energy**
equal to E  Fd . Hooke’s Law says that the force exerted by a spring displaced by a distance x
from its resting length is equal to F  kx , where k is a constant that depends on the particular
spring. In this activity you will approximate the energy required to stretch the spring with
k  0.155 N/cm from 5 cm past its natural length to 10 cm.
*
The standard unit of force is Newtons (N), where 1 N = 1 kg·m/s2 or the force required to
accelerate a 1 kg mass at 1 m/s2. Increasing either the mass or the acceleration rate therefore
requires a proportional increase in force.
**
The standard unit of energy is Joules (J), where 1 J = 1 N·m or the energy required to
move an object with a constant force of 1 N a distance of 1 m. Increasing either the force or
the distance requires a proportional increase in energy.
Lab Preparation:
1. Draw and label a large picture of a spring initially displaced 5 cm from its natural length then
stretched to a displacement of 10 cm.
2. Does it take less, the same, or more energy to stretch the spring from 7.5 cm to 10 cm than it
takes to stretch the spring from 5 cm to 7.5 cm? Explain.
3. Explain why we cannot just multiply a force times a distance to compute the energy.
Lab:
1. Draw and label a large picture of a spring initially displaced 5 cm from its natural length then
stretched to a displacement of 10 cm.
2. Explain why we cannot just multiply a force times a distance to compute the energy.
3. Use Riemann sums with 10 terms to find both an underestimate and overestimate for the energy
required to stretch the spring from 5 cm to 10 cm. Write out your sums numerically and with
summation notation. Illustrate the terms of your sum on your picture.
4. Write an algebraic expression for your error. What is the bound on the error for your
approximations? What is the range of possible values for the energy (in N·m) required to stretch
the spring from 5 cm to 10 cm?
5. Illustrate your approximations using 10 terms in terms of area under an appropriate graph. Label
your axes, function, underestimate, overestimate, actual value, error, and error bound in your
diagram.
© CLEAR Calculus 2010
Calculus 1
Fall 2013
Lab 12: Definite Integrals – Part 1
Work with your group on the context assigned to you. We encourage you to collaborate both in and
out of class, but you must write up your responses individually.
Context 2: A uniform pressure P** applied across a surface area A creates a total force* of F  PA .
The density of water is 1000 kg per cubic meter, so that under water the pressure varies according to
depth, d, as P  9800d . In this activity you will approximate the total force of the water exerted on
a dam 63.22 meters wide and extending 25 meters under water.
*
The standard unit of force is Newtons (N), where 1 N = 1 kg·m/s2 or the force required to
accelerate a 1 kg mass at 1 m/s2. Increasing either the mass or the acceleration rate therefore
requires a proportional increase in force.
Pressure is the force per unit area, P  F A , so for example a force of 6 N applied over a
2 m2 area would generate a pressure of 3 N/m2. Increasing the force would increase the
pressure proportionally. Increasing the area would decrease the pressure proportionally (an
inverse proportion).
**
Lab Preparation:
1. Draw and label a large picture of a dam 63.22 m wide and extending 25 m under water.
2. Is there less, the same, or more force on the top half of the dam or the bottom half? Explain.
3. Explain why we cannot just multiply a pressure times an area to compute the force.
Lab:
1. Draw and label a large picture of a dam 63.22 m wide and extending 25 m under water.
2. Explain why we cannot just multiply a pressure times an area to compute the force.
3. Use a Riemann sum with 5 terms to find both an underestimate and overestimate for the total
force of the water exerted on this dam. Write out your sums numerically and with summation
notation. Illustrate the terms of your sum on your picture.
4. What is the error bound for each of these approximations?
5. Illustrate your approximations using 5 terms in terms of area under an appropriate graph. Label
your axes, function, underestimate, overestimate, actual value, error, and error bound in your
diagram.
© CLEAR Calculus 2010
Calculus 1
Fall 2013
Lab 12: Definite Integrals – Part 1
Work with your group on the context assigned to you. We encourage you to collaborate both in and
out of class, but you must write up your responses individually.
Context 3: The mass M of an object with constant density d and volume v is M  dv . A 10-meter
long, 10-cm diameter pole is constructed of varying metal composition so that its density increases
at a constant rate from 4.2 grams per cubic centimeter at one end to 33.8 grams per cubic centimeter
at the other. In this activity you will approximate the mass of this pole.
Lab Preparation:
1. Draw a large picture of the pole labeling all dimensions and representing the variable density.
2. Is there less, the same, or more mass in one half of the pole than on other? Explain.
3. Explain why we cannot just multiply a density times a volume to compute the mass.
Lab:
1. Draw a large picture of the pole labeling all dimensions and representing the variable density.
2. Explain why we cannot just multiply a density times a volume to compute the mass.
3. Use a Riemann sum with 4 terms to find both an underestimate and overestimate for the mass of
the pole. Write out your sums numerically and with summation notation. Illustrate the terms of
your sum on your picture.
4. What is the error bound for each of these approximations?
5. Illustrate your approximations using 4 terms in terms of area under an appropriate graph. Label
your axes, function, underestimate, overestimate, actual value, error, and error bound in your
diagram.
© CLEAR Calculus 2010
Calculus 1
Fall 2013
Lab 12: Definite Integrals – Part 1
Work with your group on the context assigned to you. We encourage you to collaborate both in and
out of class, but you must write up your responses individually.
Context 4: The volume V of an object with constant cross-sectional surface area, A, and height, h, is
V  Ah . In this activity you will approximate the volume of water in a large spherical bottle of
radius 1 foot that is filled to height of 21.7 inches*.
*Since you can easily compute the volume of the bottom half of the sphere, you will focus on
approximating the volume contained in the remaining 9.7 inches.
Lab Preparation:
1. Draw a large picture of the spherical bottle labeling all dimensions and representing the variable
cross-sectional area at different heights.
2. Is there less, the same, or more volume in the top 4 inches of the bottle than in the second 4
inches? Explain.
3. Explain why we cannot just multiply an area times a height to compute the volume.
Lab:
1. Draw a large picture of the spherical bottle labeling all dimensions and representing the variable
cross-sectional area at different heights.
2. Explain why we cannot just multiply an area times a height to compute the volume.
3. Use a Riemann sum with 4 terms to find both an underestimate and overestimate for the volume
of water in the bottle. Write out your sums numerically and with summation notation. Illustrate
the terms of your sum on your picture.
4. What is the error bound for each of these approximations?
5. Illustrate your approximations using 4 terms in terms of area under an appropriate graph. Label
your axes, function, underestimate, overestimate, actual value, error, and error bound in your
diagram.
© CLEAR Calculus 2010
Calculus 1
Fall 2013
Lab 12: Definite Integrals – Part 1
Work with your group on the context assigned to you. We encourage you to collaborate both in and
out of class, but you must write up your responses individually.
Context 5: The average annual household income in the U.S. is $49,443 with standard deviation
$23,470. Assuming a normal distribution of household incomes, the probability density would be
f ( x) 
1
2 2
e
 ( x2)
2
2
where   49, 443 and   23, 470 . In situations where the probability density is a constant, p, the
proportion of cases falling within a range a  x  b is (b  a) p . In this activity, you will
approximate the proportion of households earning more than the mean annual income but less than
$100,000 annually.
Lab Preparation:
1. Draw a large graph of f. Show what area corresponds to the proportion of households earning
more than the mean annual income but less than $100,000 annually
2. Is there less, the same, or more of the population in the $50,000 to $75,000 income range than in
the $75,000 to $100,000 range? Explain.
3. Explain why we cannot just multiply a probability density by the size of the income range to
determine the proportion of households in that range.
Lab:
1. Draw a large graph of f. Show what area corresponds to the proportion of households earning
more than the mean annual income but less than $100,000 annually
2. Explain why we cannot just multiply a probability density by the size of the income range to
determine the proportion of households in that range.
3. Use a Riemann sum with 4 terms to find both an underestimate and overestimate for the overall
proportion. Write out your sums numerically and with summation notation. Illustrate the terms
of your sum on your picture.
4. What is the error bound for each of these approximations?
5. Illustrate your approximations using 4 terms in terms of area under an appropriate graph. Label
your axes, function, underestimate, overestimate, actual value, error, and error bound in your
diagram.
© CLEAR Calculus 2010
Calculus 1
Fall 2013
Lab 12: Definite Integrals – Part 1
Challenge Context: Fluid traveling at a velocity v across a surface area A produces a flow rate of
F  vA . Poiseuille’s law says that in a pipe of radius R, the viscosity of a fluid causes the velocity
to decrease from a maximum at the center ( r  0 ) to zero at the sides ( r  R ) according to the
 r2 
function v  vmax 1  2  . In this activity you will approximate the rate that water flows in a 4-inch
 R 
diameter pipe if vmax  4.44 ft/s.
1. Draw a large picture of a cross-section of the pipe labeling all dimensions and representing the
variable flow rate at different places.
2. Explain why we cannot just multiply a velocity times an area to compute the flow rate.
3. Use a Riemann sum with 4 terms to find both an underestimate and overestimate for the overall
flow rate in the pipe. Write out your sums numerically and with summation notation. Illustrate
the terms of your sum on your picture.
4. What is the error bound for each of these approximations?
5. Illustrate your approximations using 4 terms in terms of area under an appropriate graph. Label
your axes, function, underestimate, overestimate, actual value, error, and error bound in your
diagram.
© CLEAR Calculus 2010