Question

need help with the attachment @MIT 18.01 Single …

Answer 1

the answer choices are a) F(8) - F(1) = 31 b) F(8) - F(1) = 35 c) F(8) - F(1) = 32 d) F(8) - F(1) = 33 e) F(8) - F(1) = 34

Answer 2

By definition of the function F,
\[ F(b) - F(a) = \int_a^b f(x) \ dx \]
So what does that imply for your problem?

Answer 3

Also,
\[ F(8) - F(1) = (F(8) - F(2)) - (F(2) - F(1)) \]
\[ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 28 + \int_1^2 f(x) \ dx \]

Answer 4

However it doesn't have a function inside the integral

Answer 5

? I don't understand what you're saying.

Answer 6

the only thing they give us is f(X)

Answer 7

You have the graph of f(x) and
\[ F(8) - F(2) = \int_2^8 f(x) \ dx = 28 \]

Answer 8

And you're asked to find the integral of f(x) from 1 to 8. So you already have the integral from x = 2 to 8. Now you need to add the part from x = 1 to 2.

Answer 9

ok

Answer 10

That is,
\[ \int_1^8 f = \int_1^2 f + \int_2^8 f \]

Answer 11

You already know the second part of this. The only question then is what is the integral from 1 to 2, and you can figure that out from the graph.

Answer 12

And I'd rather not tell you what that is but ask you to figure it out.

Answer 13

From a more geometrical point of view, note that this is a piece-wise defined function where the leftmost function is clearly y=2x.\[\int_{2}^{4}2xdx=12,\]\[\int_{1}^{4}2xdx=15.\]

Answer 14

the part from 1 to 2 I counted 6 squares

Answer 15

Right, or even more geometrically, look at the area under the curve from 1 to 2 and figure out the area.

Answer 16

really? Count again.

Answer 17

4 square

Answer 18

no.

Answer 19

well there only two full complete squares between 1 and 2? Unless I'm suppose to count the area under the curve

Answer 20

Yes, what is the area under the graph of f(x) between 1 and 2.

Answer 21

If you can't figure it out geometrically, figure it out analytically,
\[ \int_1^2 2x \ dx \]

Answer 22

3

Answer 23

yes

Answer 24

Therefore
\[ F(8) - F(1) = .... what? \]

Answer 25

31

Answer 26

Yes, 28 + 3 = 31.

Answer 27

Now convince yourself geometrically that the area is 3.

Answer 28

ok, but how did you know how to figure it out analytically,

Answer 29

Because as across noted earlier, the function f(x) in that that range of x is f(x) = 2x

Answer 30

oh, ok thanks so much for helping me out James and across

Answer 31

Nobody mention the fundamental theorem of integral calculus ...

Answer 32

I didn't use those words, but in my first post, yes.

Answer 33

Okay James. btw I did found the area by looking it as a trapezium .. any other approach ?

Answer 34

aha using the equation of the straight line ..

Answer 35

Or as a rectangle and a triangle. Or as a rectangle and two segments of a square that you can put back together as one square.

Answer 36

Indeed .. cute little problem.