rest of stewart

Question 1

Riemann sums

Answer 1

sum of f(x) delta x where f is a point in that area of delta x

Question 2

definite integral

Answer 2

sum of the n riemmann sums as n aproaches infinity. each sum is of width delta x where delta x is (b-a)/n, x is then equal to a + i(b-a)/n. use summation properties

Question 3

integral is equal to

Answer 3

lim as n aproaches infinity of the sum from i=0 to n of polynomial

Question 4

midpoint rule

Answer 4

edtimation of area deltax X (F(x-) ….) where x is midpoint of the area of delta x

Question 5

That one integration identitie with m and M

Answer 5

take m and M are the largest values of f in the given interval, then m(b-a) is greater or equal to integral is greater or equal to M(b-a)

Question 6

fundamental theorem of calc

Answer 6

derivaative is the opposite of integration. if g(x) = integral f(t) and everything is continuous then g’(x) = f(x)

Question 7

ODEs

Answer 7

ordinary diff eq - only have one independent variable x and dependent variable y

Question 8

homogenous ODE

Answer 8

all terms involve dependent variable or one of its derivatives

Question 9

Separable diff eq

Answer 9

first order diff eq that can be expressed as a function of y and a function of x. dy/dx = f(x)g(y)

Question 10

separable diff eq solving

Answer 10

separate and integrate individally

Question 11

first order linear

Answer 11

in the form of dy/dx + P(x)y = Q(x)

Question 12

first order linear solving

Answer 12

find I(x) = e^int(P(x)), then d/dx(I(x)y = I(x)Q(x)

Question 13

second order homogeneous with constant coefficients

Answer 13

ay’’ + by’ +cy = 0 so the functions a ,b, c are all constant.

Question 14

second order homogeneous with constant coefficients solving

Answer 14

take y = e^rx and substitute into it. Solve quadratic for r, depending on what r is the solutions for y are different

Question 15

r has 2 solutions

Answer 15

y = Ae^r1x + Be^r2x

Question 16

r has one solution

Answer 16

y = Ae^rx + Bxe^rx

Question 17

r = alpha +- beta i

Answer 17

y = e^alphax (Acos(betax) + Bsin(betax))

Question 18

secondary non homogeneous with linear coefficients solving

Answer 18

solve for yc as normal by y=e^rx substitution.
find ypi using a general substitution, and sub into diff eq to find constants.
ygeneral = yc + ypi

Question 19

improper integral type one

Answer 19

is one of the limits of the integral goes off to infinity, we can express this limit as t. Do the integral and take the limit as t approaches infinity, then see if it converges to a number or it fucks off

Question 20

improper integral type 2

Answer 20

if there is a discontinuity at b, and otherwise function is continuous from a to c, then we can add integrals from a to b and from b to c.

Question 21

comparison theorem

Answer 21

if there are two functions, one bigger than the other. If the smaller one is divergent, so is the bigger one. If the bigger one is convergent, so is the smaller one.

Question 22

PArtial fractions 101

Answer 22

for linear, separate. for repeated, one linear one square. for irreducible, ax + b nominator. make fractions have equal denominators, add and equal to nominator of full fraction. solve

Question 23

Method of cylindrical shells

Answer 23

integrates in little shells from line of rotation. to rotate about y we integrate in x using circumference (spix) X height (f) x with (dx). we can rotate about another line x=X by making the circumference 2pi(X-x). Likewise for rotating around x, we can rotate about y = Y, integrate in dy and all ts. height is also afected

Question 24

formula for arc length

Answer 24

integral of sqrt(1+dy/dx^2