Calculus: Applications of Integration flashcards

Question 1

What is the disk method formula for volume along the x-axis?

Accepted Answer

V = ∫ from a to b of π[f(x)]^2 dx

Question 2

What is the washer method formula for volume?

Accepted Answer

V = ∫ from a to b of π[(f(x))^2 − (g(x))^2] dx

Question 3

What is the cylindrical shells formula for volume?

Accepted Answer

V = ∫ from a to b of 2πx f(x) dx

Question 4

What integral gives the arc length of y = f(x) from a to b?

Accepted Answer

∫ from a to b of √(1 + [f′(x)]^2) dx

Question 5

What is a cross-section of a solid?

Accepted Answer

The intersection of a plane with the solid

Question 6

What is a solid of revolution?

Accepted Answer

A solid made by revolving a plane region around a line in that plane

Question 7

What is the centroid of a region?

Accepted Answer

Its geometric center

Question 8

What does the symmetry principle say about a region symmetric about a line l?

Accepted Answer

Its centroid lies on l.

Question 9

By the theorem of Pappus, what is the volume of a solid made by rotating a plane region about an axis outside it?

Accepted Answer

The area of the region times the distance traveled by its centroid

Question 10

What is the moment about the origin of n point masses m_i at positions x_i on a line?

Accepted Answer

M = Σ m_i x_i

Question 11

How is the x-coordinate of a lamina’s center of mass found?

Accepted Answer

x̄ = M_y/m

Question 12

What integral gives the work done by a force F(x) from a to b?

Accepted Answer

W = ∫ from a to b of F(x) dx

Question 13

How is the natural logarithm defined as an integral?

Accepted Answer

ln x = ∫ from 1 to x of (1/t) dt

Question 14

How is e^x defined from the natural logarithm?

Accepted Answer

As the inverse function of ln x

Question 15

What model do exponential growth systems follow?

Accepted Answer

y = y0·e^(kt)

Question 16

What differential equation says the rate of growth is proportional to the quantity present?

Accepted Answer

y′ = ky

Question 17

What is the doubling time of an exponentially growing quantity?

Accepted Answer

(ln 2)/k

Question 18

What model do exponential decay systems follow?

Accepted Answer

y = y0·e^(−kt)

Question 19

What is the half-life of an exponentially decaying quantity?

Accepted Answer

(ln 2)/k

Question 20

What function describes the shape of a catenary?

Accepted Answer

y = a cosh(x/a)

The cards