Question

Consider the function. Use the notion of signed area to evaluate the following definite integral (-4,4) f(x) dx.

Answer 1

2 – 2 pi
4 – pi
2 – 4 pi
4 – 4 pi
4 – 2 pi

Answer 2

@Zarkon

Answer 3

@jim_thompson5910

Answer 4

how far did you get into this?

Answer 5

I'm not sure where to start.

Answer 6

why not break it up into 3 pieces and find the area of each piece

Answer 7

like so

Answer 8

-x-2=-16
-sqrt(4-x^2) = -6.283 - 8.588i
x-2 = -16

Answer 9

idk how you're getting that

Answer 10

Im taking the integral of (-4,4) and filling in those values for f(x).

Answer 11

i see

Answer 12

break it up into 3 pieces

Answer 13

first piece

\[\Large \int_{-4}^{-2}(-x-2)dx\]

Answer 14

second piece

\[\Large \int_{-2}^{0}-\sqrt{4-x^2}dx\]

Answer 15

-10

Answer 16

third piece

\[\Large \int_{2}^{4}(x-2)dx\]

Answer 17

evaluate each integral

Answer 18

-10
-pi
2

Answer 19

exactly one of those that you listed is incorrect (the other two are correct)

Answer 20

The first one is -2. Forgot the -x.

Answer 21

hint:

areas above the x axis are positive areas
areas below the x axis are negative areas

you can simply look and determine the sign of the area without having to do any math at all

Answer 22

So is the answer: 2-2pi?

Answer 23

Or am I jumping to conclusions?

Answer 24

list the areas of each piece again

Answer 25

-2
-pi
2

Answer 26

the -2 is incorrect

why? because the red triangle is ABOVE the x axis, so it has to be positive at least. It's some positive number. Try again

Answer 27

2
-pi
2

Answer 28

yep, now we add:

(Area1)+(Area2)+(Area3) = (2) + (-pi) + (2) = 4 - pi

Answer 29

Thanks. I was just making small mistakes, but I think I understand it now.

Answer 30

Are you able to help me through another one?

Answer 31

sure, just one more though

Answer 32

Alright thanks! I think I mainly need help breaking it into pieces.

Answer 33

The following graph of f(x) consists of line segments and semicircles. Use the graph and the notion of signed area to evaluate the definite integral (1,9) f(x) dx.

Answer 34

18 – 4pi
16 – 2.5pi
16 – 2pi
14 – pi
14 – 4pi

Answer 35

break it up like this

Answer 36

actually, you don't need one line in there, here's a more simpler version

Answer 37

Now how do I find the integrals of all those pieces?

Answer 38

well you can use plain old geometry instead of calculus to find the area of each piece

Answer 39

So I think the first piece's area is pi/2= 1.57

Answer 40

I used the area of a semicircle which is (pi x r^2)/2.(pi x 1^2)/2 =pi/2

Answer 41

I then broke the second piece into two and made a square and a triangle.

Answer 42

The area of the square is 4.

Answer 43

The area of the triangle is (4 x 2)/2=4

Answer 44

So the area of the whole second piece is 8.

Answer 45

The area of the third piece is (6x2)/2 = 6

Answer 46

The area of the last piece is (pi x 3^2)/2 = (pi x 9)/2 = 14.13

Answer 47

All together the area of all the pieces is 1.57 + 8 + 6 + 14.13 = 29.7

Answer 48

one sec

Answer 49

you forgot about the 2x2 square that is below the first semicircle

Answer 50

So then it's 33.7?

Answer 51

Wait I need to minus the last semicircle.

Answer 52

So i get 5.44.

1.57 + 4 + 8 +6+ + (-14.13).

Answer 53

I don't think that is one of my answers @jim_thompson5910

Answer 54

Did I do something wrong?

Answer 55

let me see

Answer 56

ok, two things

1) keep everything in terms of pi (don't approximate pi)

2) the semicircle under the curve has a negative area

Answer 57

So the first semicircle is pi/2 and the second semicircle is -9pi/2. I think everything else should be the same.

Answer 58

Should I multiple all the values by 2?

Answer 59

why multiply by 2?

Answer 60

So I get pi and -9pi.

Answer 61

I'd get 36 -8pi = 4-2pi.

Answer 62

or 18 - 4pi. Which is one of my answers.

Answer 63

Yeah I think that's right.

Answer 64

@jim_thompson5910

Answer 65

18-4pi is correct

Answer 66

Alright I have one more. I'm almost possible I know the answer, but do you mind if I just double check with you?

Answer 67

It's pretty basic.

Answer 68

The graph of f(x) is shown below. Which of the following are true?

Answer 69

I only.
II only.
III only.
II and III only.
I and II only. <---- My answer

Answer 70

one sec

Answer 71

I agree with what you got. Nice job

Answer 72

Thank you very much! I really appreciate you spending all this time helping me. I'd give you more medals if I could, but I can't. At the very least you have a fan.

Answer 73

you're welcome