Question

6 calc questions from homework problems?
Area and Definite Integrals?

Answer 1

here are the first 3

Answer 2

for the first one i got d

Answer 3

\[\sum_{k=3}^5 f(k)=f(3)+f(4)+f(5)\]

Answer 4

the sum of three fractions in the first one should give you d

Answer 5

yep ok number 2?

Answer 6

in the next one find the area of each rectangle...then add

like look at the first graph, find the area of each rectangle in the first graph... then just add those rectangles

then do the same for the second graph

Answer 7

ok give me a second

Answer 8

I got C. What did you get?

Answer 9

how did you get C

Answer 10

here are the other 3.

For 1. D
2.
3.
4.
5. B
6. A

Answer 11

did you find the base length of each rectangle?

Answer 12

I tried to but I'm kind of confused.

Answer 13

f(ci) is the height

Answer 14

the area lies between x=1 and x=2
and the length from x=1 to x=2 is 1
and you want to divide 1 into 5 parts
which means the base length is 1/5 for each rectangle

Answer 15

ok so the second one is 1/5 also but the heights are different.

Answer 16

right you will have to use the function given to find the the heights

Answer 17

so plug in 1/5 into the equation?

Answer 18

you will need to evaluate the function at each left endpoint of the sub-intervals
and you will need to evaluate the function at each right endpoint of the sub-intervals

Answer 19

ok so how do i do that?

Answer 20

the function given is f(x)=1/x

Answer 21

right

Answer 22

use the length of each sub-interval to determine the left and right endpoints

Answer 23

so about (1,1) and (2,0.6) ?

Answer 24

for example the first sub-interval is [1,(1+1/5)]
or [1,6/5]
the left endpoint here is x=1
the right endpoint here is x=6/5
you have 4 more sub-intervals

Answer 25

how do i find them.... sorry i am new to this.

Answer 26

the length of each sub-interval is 1/5....we found this number earlier

Answer 27

ok

Answer 28

do you know how to compute 6/5+1/5 this will give you the next right endpoint

Answer 29

7/5 or 1 2/5

Answer 30

and then 7/5+1/5= 8/5 or 1 3/5?

Answer 31

first sub-interval is [1,6/5]
second sub-interval is [6/5,7/5]
count to find the other 3 sub-intervals

Answer 32

yep the next right endpoint would be 8/5

Answer 33

[7/5.8/5] [8/5,9/5] and [9/5,10/5]

Answer 34

right and 10/5 is just 2
first sub-interval is [1,6/5]
second sub-interval is [6/5,7/5]
third sub-interval is [7/5,8/5]
fourth sub-interval is [8/5,9/5]
fifth sub-interval is [9/5,2]

Answer 35

so you take all of those left endpoints plug them into f(x)=1/x to find the height of each rectangle in the 2nd graph

and you take all of those right endpoints plug them into f(x)=1/x to find the height of each rectangle in the 1st graph

Answer 36

ok do you want me to to that or do you want to?

Answer 37

you would probably be best to do it since this is your assignment :p

Answer 38

so then find the area of all of them and add them to get our answer?

Answer 39

when i plug it into my calc it says error...

Answer 40

the area of each rectangle is base*height
where all the bases are equal
so you can just add up all the heights and then multiply the sum by the base length of find the sum of all the areas

Answer 41

when i try to plug in for the heights it says error....

Answer 42

can you give me the decimals and ill multiply it by the base since my calc isnt doing it?

Answer 43

can't tell you why it says that
1/1
and
1/(6/5)
and
1/(7/5)
and
1/(8/5)
and
1/(9/5)
are all numbers

Answer 44

ohhhhh i was using 1/[7/5,7/6] lol give me a sec

Answer 45

that is the sub-intervals....

Answer 46

the list I just gave you was left endpoints of each of those sub-intervals we mentioned

the right endpoints would exclude 1/1 and include 1/2 using that list I just mentioned

Answer 47

i got 1
5/7
5/8
5/9

so add them and multiply it by 2 to find the area?

Answer 48

what happened to 5/6
and also why are you multiplying by 2
2 is not the base length of each rectangle

Answer 49

its 1/5 for each

Answer 50

right

Answer 51

1879/504 for adding them together....... now multiply it by 1/5?

Answer 52

if so i got .74

Answer 53

0.7456349

Answer 54

that matches the upper sum of B. so it would be b?

Answer 55

\[\text{ area of rectangle }=\text{ base } \cdot \text{ height } \\ \\ \text{ the approximated area using } \\ \text{ \left endpoints } \\ \text{ recall the function is } f(x)=\frac{1}{x} \\ ... \\ \text{ so we have } \frac{1}{5} \cdot \frac{1}{1}+\frac{1}{5} \cdot \frac{1}{\frac{6}{5}}+\frac{1}{5} \cdot \frac{1}{\frac{7}{5}}+\frac{1}{5} \cdot \frac{1}{\frac{8}{5}}+\frac{1}{5} \cdot \frac{1}{\frac{9}{5}} \\ \text{ or } \frac{1}{5}(1+\frac{5}{6}+\frac{5}{7}+\frac{5}{8}+\frac{9}{5}) \approx .75\]

Answer 56

made a type-o
5/9 not 9/5

Answer 57

\[\text{ \right endpoint rule for this problem looks like } \\ \frac{1}{5}(\frac{5}{6}+\frac{5}{7}+\frac{5}{8}+\frac{5}{9}+\frac{1}{2})\]

Answer 58

so would it be b then?

Answer 59

if you want to verify it is b then evaluate the right endpoint rule thing I just wrote

Answer 60

but yep you should come out with that is b
because the only choice that has the answer .75 is b

Answer 61

ok

Answer 62

can you check my other answers before we move to #3?

Answer 63

1D (we did)
2 b (we did)
3
4
5 B
6 A

Answer 64

5 and 6 appear to be right

Answer 65

ok now i just need help with 3 and 4 :)

Answer 66

this number 3 is similar to what we did
just the graph isn't given this time

Answer 67

well and except it isn't finite amount of rectangles

Answer 68

and I suppose you could use a definite integral instead of using the whole summing thing...

Answer 69

the area between y=f(x) and y=0 on the interval from x=a to x=b can be found be evaluating
\[\int\limits_a^b f(x) dx\]

Answer 70

here's the graph

Answer 71

dan already straight out gave the answer to number 5 earlier

Answer 72

oops number 4

Answer 73

oh i must not have seen it give me a sec

Answer 74

well we also said you find the height by using the...

Answer 75

left endpoints and sub integrals

Answer 76

not alone
you have to plug them into the...

Answer 77

f(x) equation

Answer 78

yes
c_i represents the sample point we are choosing in the sub-interval
so f(c_i) represents the height we are using for each sub-interval

Answer 79

this is for #3 correct?

Answer 80

which ever one was about determining which one is the height

Answer 81

oh ok that's #4

Answer 82

oh i get it now.... ok last one number 3 finding the area of the bonded region

Answer 83

have you used what I said for number 3 yet?
or did you want to find the answer using the proper way?
the proper was being the way they want you to find it?

Answer 84

i posted the graph where do i go from there?

Answer 85

if you want post the steps out and ill follow along according to the other problem and find the final answer.....

Answer 86

ok we will go with the limits thing intead of the definite integral I suggested above
\[\Delta x \cdot f(a+1 \Delta x)+\Delta x \cdot f(a+2 \Delta x)+\Delta x \cdot f(a+3 \Delta x)+ \cdots \Delta x \cdot f(a+n \Delta x) \\ =\Delta x \cdot [ f(a+1 \Delta x)+f(2+\Delta x)+ f(3+ \Delta x)+ \cdots +f(a+n \Delta x)] \\ =\Delta x \cdot \sum_{i=1}^n f(a+i \Delta x) \\ \\ \text{ so } \int\limits_a^b f(x) dx=\lim_{x \rightarrow \infty} \Delta x \cdot \sum_{i=1}^{n } f(x+i \Delta x)=Area\]

Answer 87

the limit way is the way they want you to go about it

Answer 88

the integral is another way you go about it if you had a choice

Answer 89

is this still for #4? I am talking about number 3. I understand #4

Answer 90

we are on number 3
number 4 was a short answer

Answer 91

ok want we me open a new question with that one and tag u?

Answer 92

well I kinda wrote everything you need for number 3 in this question ...

Answer 93

oh ok nvm then we can continue.