Thread: confusion with a question about integrals, multiple choices

Hi.

Given

$\displaystyle \int_{-1} ^2 {f(x) =a}$ and $\displaystyle \int_0 ^2 {f(x) = b $

the value of
$\displaystyle \int_0 ^{-1} f(x) $ is:

A. a+b
B. 2a-b
C. a-b (which I think is right)
D. a-2b
E. b-a (might be true?)

I am confused because the last integral boundaries are from 0 to -1 and I always see them written from the smaller to the bigger number, aka from -1 to 0.
Does this makes a the result different? (maybe it's a mistake in the printing?)

Thanks.


attached image is another question, sorry for the unprofessional editing, it's from the book and it's not in English so i had to edit it a little... the pen markings are my own, what I think are correct/incorrect.
So I basically think statements 2 and 3 are correct, therefore answer D would be my choice, what do you think?

I need to send this out today and have total of 18 problems to answer (luckily this time they are multiple choices, not "open ended")
I would probably add more pictures as it's much easier and faster than typing the whole equations in the forum...

Thanks.

Originally Posted by ryu1

Given
$\displaystyle \int_{-1} ^2 {f(x) =a}$ and $\displaystyle \int_0 ^2 {f(x) = b $
the value of
$\displaystyle \int_0 ^{-1} f(x) $ is:

$\displaystyle \int_{ - 1}^0 {f} + \int_0^2 {f} = \int_{ - 1}^2 {f} $ and $\displaystyle \int_{ - 1}^0 {f} = - \int_0^{ - 1} {f} $

SO?

Originally Posted by Plato

$\displaystyle \int_{ - 1}^0 {f} + \int_0^2 {f} = \int_{ - 1}^2 {f} $ and $\displaystyle \int_{ - 1}^0 {f} = - \int_0^{ - 1} {f} $

SO?

so it's E. b-a

$\displaystyle \int_{ - 1}^0 {f} [= z] + \int_0^2 {f} [= b] = \int_{ - 1}^2 {f} [= a] $

z+b=a
z=a-b
-z=b-a

right?

Originally Posted by ryu1

so it's E. b-a

$\displaystyle \int_{ - 1}^0 {f} (= z) + \int_0^2 {f} (= b) = \int_{ - 1}^2 {f} (= a) $

z+b=a
z=a-b
-z=b-a

right?

Correct!

Originally Posted by Plato

Correct!

Thank you very much!

What you think about the problem in the picture?

Thanks.

Hello, ryu1!

Attached image is another question.
So I basically think statements 2 and 3 are correct.
Therefore, answer D would be my choice. .What do you think? . I agree!

Code:

      |
      *
      |   *
      |   |::*
      |   |::::*      4 
    --+---+------*----+----
      |   1      3*:::|
      |             *:|
      |               *

The graph of $\displaystyle f(x)$ is continuous and passes through $\displaystyle (3,0).$
$\displaystyle S$ = shaded area.
$\displaystyle F(x)$ is the antiderivative of $\displaystyle f(x).$

$\displaystyle 1.\;S \:=\:\int^4_1 f(x)\,dx \qquad\qquad\qquad\qquad 2.\;\int^4_1f(x)\,dx \:=\:F(4) - F(1) $

$\displaystyle 3.\;S \:=\:\int^3_1f(x)\,dx - \int^4_3f(x)\,dx \qquad 4.\;S \:=\:F(4) - F(1) $


Which are the correct statements?

$\displaystyle \begin{array}{cccccccccccc}(A)& 2 && (B) & 1,3 && (C) & 2,4 && (D) & 2,3 \\ \\ (E)&1,2,4 && (F)&1 && (G) &1,2,3 && (H) & 3 \end{array}$

Statement 1 is incorrect.
The integral from 1 to 4 is not the shaded area.
It is the net area (some of which is negative).

Statement 4 is equivalent to statement 1.
It is also incorrect.

Statement 2 is correct.
It gives the value of the definite integral.
$\displaystyle \int^4_1f(x)\,dx \:=\:F(x)\bigg]^4_1 \:=\:F(4)-F(1)$

Statement 3 is correct.
The total shaded area is the integral from 1 to 3
. . minus the integral from 3 to 4.

Originally Posted by Soroban

Hello, ryu1!



Statement 1 is incorrect.
The integral from 1 to 4 is not the shaded area.
It is the net area (some of which is negative).

Statement 4 is equivalent to statement 1.
It is also incorrect.

Statement 2 is correct.
It gives the value of the definite integral.
$\displaystyle \int^4_1f(x)\,dx \:=\:F(x)\bigg]^4_1 \:=\:F(4)-F(1)$

Statement 3 is correct.
The total shaded area is the integral from 1 to 3
. . minus the integral from 3 to 4.

Thank you.

New problem:
Given the function , R is a constant.

Find the anti derivative F(x) that such that F(R) = 0.

What is the value of F(0.5R)+F(2R)+F'(0.5R)+F'(2R)?

Answers:


I tend to go with C because I somehow got to 3/2R after calculating only the F'(0.5R)+F'(2R) (these are just the given function right?)
But I have a feeling the F(0.5R)+F(2R) will change it to something else...also where does the 0.19 comes from I wonder.

Thanks