Area B/t Curves
Volumes of Rev.
Arc Length
Surface Area
Diff. Eq.
100
Graph the functions: e^x, x^2, lnx, 1/x
y=e^x:
y=x^2:
y=ln(x):
y=1/x:
100
What are the equations for Disk/Washer method, and Cylindrical Shells method?
Disk/washer:
V=int_a^bpi[r_(out)^2-r_(i n)^2]dx
V=int_c^dpi[r_(out)^2-r_(i n)^2]dy
Cylindrical Shells:
V=int_a^b2pirhdx
V=int_c^d2pirhdy
Where r=distance from aor, and h=distance b/t graphs
100
What is the equation for Arc Length?
L=int_a^bsqrt(1+(dy/dx)^2)dx
L=int_c^dsqrt(1+(dx/dy)^2)dy
100
What is the equation for Surface area?
SA=2piint_a^br*sqrt(1+(dy/dx)^2)dx
SA=2piint_c^dr*sqrt(1+(dx/dy)^2)dy
Where r=distance from aor
100
How do you solve a separable differential equation?
Possible answer:
Isolate all x's (and the dx) on one side of the equation and all y's (and dy) on the other side. Then integrate both sides. Then solve for y to get the general solutio.
For particular solution, plug in the given point and solve for C. Then plug C back into the General solution.
200
What is area between curves and how do you find it?
Possible answer: Area between curves is the area between two different curves or functions. You find it by taking the area of the bigger function minus the area of the smaller one. Which is the same as the integral of the bigger function minus the integral of the smaller.
200
Find volume for y=2x and y=x^2 with AOR:x-axis
64/15pi
200
Find length:
y=1/6 (x^2+4)^(3/2), 0<=x<=3
15/2
200
Find Surface Area for AOR:x-axis:
y=x^3
, [0,2]
pi/27(145sqrt(145)-1)
200
Find general Solution:
2ye^(y^2)y'=2x+3x^(1/2)
+-sqrtln(x^2+2x^(3/2)+C)
300
Find area between curves for y=1/x and y=x^2, y=0, x=e
4/3
300
x=1+y^2 and y=x-3 with AOR: y-axis
117/5
300
Set up the integral to find the arclength of
y=2x^2+6x-3
from [-2,4]
int_(-2)^4 sqrt(1+(4x+6)^2)
300
Set up integral to find Surface Area for
AOR: x-axis:
y=tanx
, from
x=[0,pi/3]
int_0^(pi/3) 2pitanx*sqrt(1+sec^4x)dx
300
Find General Solution:
dx/dt=1-t+x-tx
x=-1+Ce^(t-(t^2)/2)
400
Find area b/t y=x^2 and y=4x-x^2
8/3
400
Find Volume: x=0, x=9-y^2 with AOR x=-1
256pi
400
Set up the integral to find the arclength of
x=y^(1/2)-y
y=[1,4]
int_1^4 sqrt(1+(1/(2sqrt(y))-1)^2)
400
Set up integral to find Surface Area for, AOR: y-axis:
y=x^(-2)
, x=[1,2]
int_1^2 2pi*x*sqrt(1+4x^(-6))dx
400
Solve the IVP:
(dr)/dt+2tr=r
,
r(0)=5
r(t)=5e^(t-t^2)
500
Find the area between x+y=0 and x=y^2+3y
32/3
500
Set up: y=(x)^1/2, y=x^2 with AOR y=2
int_0^1 pi[(2-x^2)^2-(2-sqrt(x))^2]
500
Set up and evaluate the integral to find the arclength for
y=2ln(sin(1/2x))
, from
[pi/3,pi]
2ln|2-(3)^(1/2)|
500
Set up integral for surface area for
y=root3(x)
rotated around y=-1 from y=[1,2]
int_1^2 2pi[(y^3-2)*sqrt(1+9y^4)]dy
500
Solve the IVP
(1+cos(x))y'=(1+e^-y)sin(x)
,
y(0)=0
ln((3-cosx)/(1+cosx))