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24 Cards in this Set
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a = ∫(f(x)g(x),x,a,b)

Areas Between Curves
This equation finds the area of a curve bounded by a and b and above by f(x), below by g(x) 

a = ∫(g(t)f˙(t),t,c,d)

Areas Between Curves
This equation is used for parametric curves. In the equation x = f(t) and y = g(t) 

v = ∫(A(x),x,a,b)

Volumes
This equation finds a volume with respect to x. This means that x is sliced perpendicular. 

v = ∫(A(y),y,c,d)

Volumes
This equation finds a volume with respect to y. This means that y is sliced perpendicular. 

A = π(r)^2

Volumes
The volume of a disk 

A = π(outer radius)^2  π(inner radius)^2

Volumes
The volume of a washer 

r^2 = (xh)^2 + (yk)^2

Volumes
The equation of a circle 

y = kx^2

Volumes
The equation of a parabola 

L = ∫(√((dx/dt)^2 + (dy/dt)^2)),t,a,b)

Arc Length
This is the equation to find the length of a parametric curve between a and b with x = f(t) and y = g(t) 

L = ∫(√(1=(dy/dx)^2),x,a,b)

Arc Length
The is the equation to find the length of a curve with given y = f(x). Regard x as a parameter and set the parametric equation equal to x=x and y=f(x) 

L = ∫(√((dx/dy)^2+1),y,a,b)

Arc Length
This equation is used when finding the length given x = f(y). 

Fave = (1/(ba) ∫(f(x),x,a,b)

Average Value
(1) This finds a average value of a function between a and b. It uses the mean value theorem. The function must be continuos. 

∫(f(x),x,a,b) = f(c)(ba)

Average Value
(2) This finds a average value of a function between a and b. It uses the mean value theorem. The function must be continuos. 

w = lim(n>inf)∑(f(x)∆x,i=1,n) = ∫(f(x),x,a,b)

Applications to Physics and Eng.
This equation is used to find the work moving a object from a to b 

f(x) = kx

Applications to Physics and Eng.
This is called Hookes Law. Y = newtons, x=displacement and k = spring constant (different for each spring). The spring function can be integrated to find work from a distance. SI unit is ftlb. 

My = ∂∫(x*f(x),x,a,b)

Applications to Physics and Eng.
This equation finds the moment about the y axis. 

Mx = ∂∫((1/2)f(x)^2,x,a,b)

Applications to Physics and Eng.
This equation finds the moment about the x axis. 

m(x hat) = My

Applications to Physics and Eng.
This equation sets a basic relationship between the mass, the x coordinate of the centroid and the moment around the _ axis 

m(y hat) = Mx

Applications to Physics and Eng.
This equation sets a basic relationship between the mass, the y coordinate of the centroid and the moment around the _ axis 

(x hat) = My/m

Applications to Physics and Eng.
This equation is a way to solve for the x coordinate of the centroid. 

(y hat) = Mx/m

Applications to Physics and Eng.
This equation gives a way to solve for the y coordinate of the centroid using M_ and m. 

m = ∂A = ∂∫f(x),x,a,b)

Applications to Physics and Eng.
This is the equation (multiple equations) to find the mass of a area. 

(x hat) = 1/A * ∫(xf(x),x,a,b)

Applications to Physics and Eng.
This equation finds the x coordinate of the centroid. 

(y hat) = 1/A * ∫((1/2)f(x)^2,x,a,b)

Applications to Physics and Eng.
This equation finds the y coordinate of the centroid. 