Integral Applications Flashcards
Cross Sections
-The volume would be the sum of the cross sectional areas times _.
Height
Mean Value Theorem for Integrals
- f(x) is continuos on [a,b]
- Then there exists some member c such that a <= c <= b and fâ(c) = [f(b) - f(a)] / (b - a) = _ = [S(ab)fâ(x)dx] / (b - a) = _
Slope, average
_ Method
- A = pi( f(x) )^2 - pi( g(x) )^2
- V = S(ab) pi( f(x)^2 - g(x)^2 )dx
Washer
Area of Isosceles Triangle
(1/4)h^2
Cross Sectional Area Perpendicular to Y Axis
- =
- _ = _1 , _2
- _ - _
- S(_12) V( _ - _ )d
y, right, left
Cross Sectional Area Perpendicular to X Axis
- =
- _ = _1 , _2
- _ - _
- S(_12) V( _ - _ )d
x, top, bottom
Motion
- Sa(t)dt = _ = antiderivative (+c)
- Plug in numbers where applicable
- Sv(t)dt = _ = antiderivative (+c)
- Plug in numbers where applicable
v(t), s(t)
Vertical Disk
- around x=
- S(_12) [ pi ( _ - _ )^2 ]d
y, right, left
Horizontal Disk
- around y=
- S(_12) [ pi ( _ - _ )^2 ]d
x, top, bottom
Disk X Axis
- =
- _ = _1 , _2
- V = pir^2h
- S(_12)[ pi ( _ - _ )^2 ]d
x, top, bottom
Disk Y Axis
- =
- _ = _1 , _2
- V = pir^2h
- S(_12)[ pi ( _ - _ )^2 ]d
y, right, left
Area of Equilateral Triangle
(sqt(3)/4)h^2
Three Line Area
-area1 + area 2 = _ + _
S(ab)f(x)dx, S(bc)f(x)dx
Displacement
- Velocity
- _
S(t1t2)v(t)dt
Distance
- Speed
- _
S(t1t2) | v(t) | dt
Positition
- t1 + S(t1t2)v(t)dt
- _ + _
Value, change
Rate
- S(t1t2)r(t)dt = _ _ = antiderivative
- Change in what rate _ not the rate itself
- Known as _
Net change, measures, accumulation
Average Value
- (Height)(width) = _
- favg(b - a) = _
- Closed interval means include the endpoints
Area, S(ab)f(x)dx
Average Acceleration
- aavg over [a,b]
- aavg = _
- antiderivative
1/(b-a) S(ab) a(t)dt
Horizontal Washer
- y=
- A = pi(ro)^2 - pi(r-)^2
- ro= _ - _
- ri= _ - _
- =
- _=_1_2
- S(_12) pi[ (ro)^2 - (ri)^2 ]d
Top, bottom, x
Vertical Washer
- x=
- A = pi(ro)^2 - pi(r-)^2
- ro= _ - _
- ri= _ - _
- =
- _=_1_2
- S(_12) pi[ (ro)^2 - (ri)^2 ]d
Right, left, y
Vertical Area
- =
- S(_12) [ f(x) - g(x) ]d
- _ - _
x, top, bottom
Horizontal Area
- =
- S(_12) [ f(x) - g(x) ]d
- _ - _
y, right, left
Area f(x)
S(x1x2)f(x)dx
Area f(y)
S(y1y2)f(y)dy
Composite Area
-Slit up integrals based where top/bottom or right/left change and _
Sum
Graphs
- Negative means faces _
- Check values in between x^3
- y=x^2
- y=x^3
- y=sqt(x)
- y^2=x
Opposite
