How many people?
How many people have walked on the Moon? [within 3]
What is 12?
How to set up a double integral in polar coordinates
What is: express f(x,y) as f(r cos(theta), r sin(theta)) and use r dr dtheta as the "area element" ?
D is the region between the graphs of 2 functions y = g1(x) and y = g2(x)
What is vertically simple?
The "area element" dA in polar coordinates
What is r dr d theta ?
1/(area)int int_D f(x,y) dA
What is the average value of f?
How many Americans are homeless? [within 10%]
What is 653,104?
How to set up a triple integral in spherical coordinates
What is: express f(x,y,z) as f(rho sin phi cos theta, rho sin phi sin theta, rho cos phi) and the "volume element" as rho^2 sin phi d rho d theta d phi ?
D is the region between the 2 graphs x = g1(y) and x = g2(y)
What is horizontally simple?
The "volume element" dV in cylindrical coordinates
What is r dz dr d theta ?
1/(volume)int int int_W f(x,y,z) dV
What is the average value of f?
How many billionaires are there in the world? [within 10%]
What is 2,781?
int int_D rho(x,y) dA =
where
rho(x,y)
is mass density
What is mass ?
Every ray from the origin intersects the region in a line segment such that the segment begins on r1(theta) and ends on r2(theta)
What is radially simple?
The "volume element" dV in spherical coordinates
What is rho^2 sin phi d rho d phi d theta ?
If g(x,y) represents population density (people/square mile), then int_0^5 int_0^5 g(x,y) dx dy =
What is the number of people living in those 25 square miles?
How many people and organizations have won a Nobel Prize? [within 10%]
What is 992?
1/A int int_d x dA
where
rho
is constant
What is bar x, the x-coordinate of the centroid?
Every ray from the origin intersects a solid in a line segment with endpoints on rho1(theta,phi) and rho2(theta, phi)
What is centrally simple?
int int_D 1 dA
What is area?
If p is a joint probability density function, then int_a^b int_c^d p(x,y) dy dx =
What is the probability that x is between a and b, and y is between c and d?
How many humans have lived in the past 192,000 years? [within 10%]
What is 109 billion?
(int int_D x rho(x,y) dA)/(int int_D rho(x,y) dA) =
What is the x-coordinate of the center of mass?
int_c^d int_a^b f(x,y) dy dx = int_a^b int_c^d f(x,y) dx dy
What is Fubini's Theorem?
int int int_W 1 dV
What is volume?
If p is a joint probability density function, then int_-oo^oo int_-oo^oo p(x,y) dx dy =
What is 1?