Integral Applications

Question 1

Cross Sections

-The volume would be the sum of the cross sectional areas times _.

Answer 1

Height

Question 2

Mean Value Theorem for Integrals

• f(x) is continuos on [a,b]
• Then there exists some member c such that a <= c <= b and f’(c) = [f(b) - f(a)] / (b - a) = _ = [S(ab)f’(x)dx] / (b - a) = _

Answer 2

Slope, average

Question 3

_ Method

• A = pi( f(x) )^2 - pi( g(x) )^2
• V = S(ab) pi( f(x)^2 - g(x)^2 )dx

Answer 3

Washer

Question 4

Area of Isosceles Triangle

Answer 4

(1/4)h^2

Question 5

Cross Sectional Area Perpendicular to Y Axis

• =
• _ = _1 , _2
• _ - _
• S(_12) V( _ - _ )d

Answer 5

y, right, left

Question 6

Cross Sectional Area Perpendicular to X Axis

• =
• _ = _1 , _2
• _ - _
• S(_12) V( _ - _ )d

Answer 6

x, top, bottom

Question 7

Motion

• Sa(t)dt = _ = antiderivative (+c)
• Plug in numbers where applicable
• Sv(t)dt = _ = antiderivative (+c)
• Plug in numbers where applicable

Answer 7

v(t), s(t)

Question 8

Vertical Disk

• around x=
• S(_12) [ pi ( _ - _ )^2 ]d

Answer 8

y, right, left

Question 9

Horizontal Disk

• around y=
• S(_12) [ pi ( _ - _ )^2 ]d

Answer 9

x, top, bottom

Question 10

Disk X Axis

• =
• _ = _1 , _2
• V = pir^2h
• S(_12)[ pi ( _ - _ )^2 ]d

Answer 10

x, top, bottom

Question 11

Disk Y Axis

• =
• _ = _1 , _2
• V = pir^2h
• S(_12)[ pi ( _ - _ )^2 ]d

Answer 11

y, right, left

Question 12

Area of Equilateral Triangle

Answer 12

(sqt(3)/4)h^2

Question 13

Three Line Area

-area1 + area 2 = _ + _

Answer 13

S(ab)f(x)dx, S(bc)f(x)dx

Question 14

Displacement

• Velocity
• _

Answer 14

S(t1t2)v(t)dt

Question 15

Distance

• Speed
• _

Answer 15

S(t1t2) | v(t) | dt

Question 16

Positition

• t1 + S(t1t2)v(t)dt
• _ + _

Answer 16

Value, change

Question 17

Rate

• S(t1t2)r(t)dt = _ _ = antiderivative
• Change in what rate _ not the rate itself
• Known as _

Answer 17

Net change, measures, accumulation

Question 18

Average Value

• (Height)(width) = _
• favg(b - a) = _
• Closed interval means include the endpoints

Answer 18

Area, S(ab)f(x)dx

Question 19

Average Acceleration

• aavg over [a,b]
• aavg = _
• antiderivative

Answer 19

1/(b-a) S(ab) a(t)dt

Question 20

Horizontal Washer

• y=
• A = pi(ro)^2 - pi(r-)^2
  
  • ro= _ - _
  • ri= _ - _
• =
• _=_1_2
• S(_12) pi[ (ro)^2 - (ri)^2 ]d

Answer 20

Top, bottom, x

Question 21

Vertical Washer

• x=
• A = pi(ro)^2 - pi(r-)^2
  
  • ro= _ - _
  • ri= _ - _
• =
• _=_1_2
• S(_12) pi[ (ro)^2 - (ri)^2 ]d

Answer 21

Right, left, y

Question 22

Vertical Area

• =
• S(_12) [ f(x) - g(x) ]d
• _ - _

Answer 22

x, top, bottom

Question 23

Horizontal Area

• =
• S(_12) [ f(x) - g(x) ]d
• _ - _

Answer 23

y, right, left

Question 24

Area f(x)

Answer 24

S(x1x2)f(x)dx

Question 25

Area f(y)

Answer 25

S(y1y2)f(y)dy

Question 26

Composite Area | -Slit up integrals based where top/bottom or right/left change and _

Answer 26

Sum

Question 27

Graphs - Negative means faces _ - Check values in between x^3 - y=x^2 - y=x^3 - y=sqt(x) - y^2=x

Answer 27

Opposite