Chapter Seven

Question 1

what is the formula for finding the volume of a solid using disks?

Answer 1

V = π∫_a^b r² dx

Question 2

What is the formula for finding the volume of a solid using washers?

Answer 2

V = π∫_a^b (R² - r²) dx

Question 3

What is the formula for finding the volume of a solid using shells?

Answer 3

V = 2π∫_a^b xf(x) dx

Question 4

What is the general formula for vertical position?

Answer 4

y(t) = -16t² + V₀t + y₀

Question 5

what is the area beween two curves?

Answer 5

A = ∫_a^b (f(x) - g(x)) dx

Question 6

what is the average value of a function?

Answer 6

A_v = 1/(b-a) ∫_a^b f(x) dx

Question 7

What is the formula for finding the volume using vertical cross sections? What about horizontal cross sections?

Answer 7

Vertical: V = ∫_a^b A(x) dx

Horizontal: V = ∫_a^b A(y) dy

Question 8

what is the formula for the work done with a constant force?

Answer 8

W = F(∆D)

Question 9

what is the formula for the work done with a changing force?

Answer 9

W = ∫_a^b F(x) dx

Question 10

what is Hooke’s Law?

Answer 10

the force required to stretch a spring x units beyond its resting position is equal to F(x) = kx, where k is a constant and x is the distance

Question 11

what is the work done in lifting an object?

Answer 11

W = (weight of object)(distance lifted)

Question 12

what is the work done to pump all of a substance out of a container?

Answer 12

W = ∫_a^b ∂ s(x) A(x) dx

where ∂ is the weight of the substance, s(x) is the distance the substance is to be moved, and A(x) is the volume of the substance

Question 13

what are the steps for solving work problems for pumping a substance?

Answer 13

1) find Volume_slice
2) find Force_slice = ∂v
3) find find Distance_slice
4) Work_slice = F_slice * D_slice
4) integrate W_slice, where the bounds of integration are the beginning and end distances

Question 14

what is the moment about the y-axis?

Answer 14

M_y = m₁x₁ + m₂x₂ + . . . + m_nx_n

m is the point mass, x is the x-coordinate of the point

Question 15

what is the moment about the x-axis?

Answer 15

M_x = m₁y₁ + m₂y₂ + . . . + m_ny_n

m is the mass of the point and y is the y-coordinate of the point

Question 16

what is the center of mass?

Answer 16

x bar = (M_y/m)

M_y is the moment about the y-axis, and m is the sum of all the masses

y bar = (M_x/m)

M_x is the moment about the x-axis, and m is the sum of all the masses

Question 17

how can the center of mass of a lamina be found?

Answer 17

Mx = 1/2 ∫_a^b (f(x))² dx

M_y = ∫_a^b xf(x) dx

x bar = (M_y/A)

y bar = (M_x/A)

or:

y bar = 1/2A ∫_a^b (f(x))² dx

x bar = 1/A ∫_a^b xf(x) dx

Question 18

what is the arc length formula?

Answer 18

L = ∫_a^b √(1 + (f(x))²) dx