Calc 4

Question 1

LRAM

Answer 1

rectangles left corner touches th ecurve

x: left corner x value
y: height of x value

(sum of heights)(width of each rectangle)= area

Question 2

MRAM

Answer 2

.5h(LRAM + RRAM)
middle of rectangle touches curve

usually the most accurate

Question 3

inscribed rectangles

Answer 3

all lie below curve

Question 4

circumscribed rectangles

Answer 4

all lie above the curve

Question 5

trapezoidal rule find area

Answer 5

width of subinterval/2 [f(x) + 2f(x1) + 2f(x2) + f(x)}

Question 6

indefinite integration

antiderivative=intergrand

Answer 6

F'(x)= f(x)
{f(x)dx
-
-add one to exponent
-use scalar rule
-divide by new exponent
-ADD +C

when dividing by negative exponent, whole new terms becomes negative

{1/x dx= ln[x]

Question 7

find c at a given point

Answer 7

get antiderivative then plug in

Question 8

reimann sum

Answer 8

find area of each rectangle and add it together. subinterval may be different so if X is larger than one, find MRAM

as width (X) approaches 0, # of rectangles approaches infinity

Question 9

subinterval

Answer 9

width of a rectangle

longest is [[P]]

Question 10

partition

Answer 10

entire interval

Question 11

velocity

Answer 11

area is distance traveled

Question 12

calculator

Answer 12

math 9

Question 13

definite integral

Answer 13

if continous
F(b) - F(a)
area: absolute value
displacement: substract

find antiderivative first, then plug a & b into F(X)

Question 14

definite integral theorem

Answer 14

scalar

sum/diff

Question 15

average value

Answer 15

is the area of the function is constant over that specific interval.

height of a rectangle that has the same area as under the curve

1. find definite integral n divide over the number of values btw a/b

Question 16

mean value theorem

Answer 16

1/b-a { f(x)dx = f(c )

C is the point when the y value of the curve is same as height of rectangle

Question 17

fundamental theorem of calculus

taking the derivative of the intergral

Answer 17

b= X
a= constant

f’= t instead of x
f(x)= { f’(t)dt
replace t w/ X

if b= u
then it is u’f(u)

Question 18

net area

Answer 18

finding integral. displacement

to find actual area, divide up the interval n change the negative area to positive n add to positive area

Question 19

u-sub type 1

Answer 19

define u
find u'
replace u n du 
solve for antiderivative
plug in real u

Question 20

u-sub type 2

Answer 20

contant attached to dx, move it over to du n add infront of integral

ISOLATE dx unless its in the integral

Question 21

u-sub type 3

Answer 21

can’t use product or quotient rule

define u then isolate x n plug in before getting rid of derivatives

Question 22

u-sub with definite integral

Answer 22

adjust bounds
plug in b/a into u equation to find u= new bounds

use same thing for regular u-sub. bring scalar out

Question 23

d/dx sin

Answer 23

cos

Question 24

d/dx cos

Answer 24

-sin

Question 25

d/dx tan

Answer 25

sec^2

Question 26

d/dx sec

Answer 26

sectan

Question 27

d/dx csc

Answer 27

-csccot

Question 28

d/dx cot

Answer 28

-csc^2

Question 29

to find position

Answer 29

s(a) + {v(t)dt

s(a)= starting position

Question 30

to find distance not displacement

Answer 30

figure out when v(t) is 0

divide integral up n solve for antiderivative/position

Question 31

to find max accel

Answer 31

find a’

then plug into a equation to c which value is higher

Question 32

integral
1
/ 1- x^2

Answer 32

arcsinx+c

Question 33

integral
1
1+x^2

Answer 33

arctanx +c

Question 34

integral
1
[x] /x^2-1

Answer 34

arcsecx

Question 35

trig substitution

Answer 35

set up triangle

make an equation with x/constant

take derivative of both sides

set @=something

new function that includes rest of integral

subsitite

find antiderivative

if definite then reset bounds w/ @=something

Question 36

u=asin@

Answer 36

a^-u^

Question 37

u=atan@

Answer 37

a^+u^

Question 38

u=asec@

Answer 38

u^-a^