Definite Integral of Absolute Value Function

[Saw]

I've just started studying Integrals and Definite Integrals, and got stuck. I tried applying the Power Rule, but the result doesn't match with the answers.

Here is the question:

. . .\(\displaystyle \mbox{(9) Let }\, f(x)\, =\, \big| \, x^2\, -\, 1\, \big| .\, \mbox{ Then }\, f(0)\, =\, \boxed{\,(1)\,}\, \mbox{ and }\, \)\(\displaystyle \displaystyle \int_0^2\, f(x)\, dx\, =\, \boxed{\, (2)\,}.\)

And the answers are:
1-) 1
2-) 2

How can I solve this?

Thanks in advance.

 

[Deleted member 4993]

I've just started studying Integrals and Definite Integrals, and got stuck. I tried applying the Power Rule, but the result doesn't match with the answers.

Here is the question:

. . .\(\displaystyle \mbox{(9) Let }\, f(x)\, =\, \big| \, x^2\, -\, 1\, \big| .\, \mbox{ Then }\, f(0)\, =\, \boxed{\,(1)\,}\, \mbox{ and }\, \)\(\displaystyle \displaystyle \int_0^2\, f(x)\, dx\, =\, \boxed{\, (2)\,}.\)

And the answers are:
1-) 1
2-) 2

How can I solve this?

Thanks in advance.

Solve what? What were you asked to calculate?

 

[HallsofIvy]

I've just started studying Integrals and Definite Integrals, and got stuck. I tried applying the Power Rule, but the result doesn't match with the answers.

Here is the question:

. . .\(\displaystyle \mbox{(9) Let }\, f(x)\, =\, \big| \, x^2\, -\, 1\, \big| .\, \mbox{ Then }\, f(0)\, =\, \boxed{\,(1)\,}\, \mbox{ and }\, \)\(\displaystyle \displaystyle \int_0^2\, f(x)\, dx\, =\, \boxed{\, (2)\,}.\)

And the answers are:
1-) 1
2-) 2

How can I solve this?

Thanks in advance.

Do you not know what |x| means? If so the first question, "What is |x^2- 1| when x= 0?" should be easy. If not you need to talk to your teacher.

\(\displaystyle x^2- 1= (x- 1)(x+ 1)\). For x< -1, both x- 1 and x+ 1 are negative so their product is positive. For -1< x< 1, x- 1 is negative but x+ 1 is positive so their product is negative. For x> 1, both are positive so their product is positive. Here, you are integrating from 0 to 1 so x is always between -1 and 1. So what should you use for \(\displaystyle |x^2- 1|\)?

 

[Saw]

Solve what? What were you asked to calculate?

I'd like to know how can I get to the results (more specifically the second question), and what do I have to review/study in order to know how to do so by my own. All I know is about the Power Rule, but doesn't seem to work with that question, considering it is a modular function...

Do you not know what |x| means? If so the first question, "What is |x^2- 1| when x= 0?" should be easy. If not you need to talk to your teacher.

\(\displaystyle x^2- 1= (x- 1)(x+ 1)\). For x< -1, both x- 1 and x+ 1 are negative so their product is positive. For -1< x< 1, x- 1 is negative but x+ 1 is positive so their product is negative. For x> 1, both are positive so their product is positive. Here, you are integrating from 0 to 1 so x is always between -1 and 1. So what should you use for \(\displaystyle |x^2- 1|\)?

Yes, I know what |x| means. What I want to know is how to get to the 2 result with the Definite Integral calculus, considering it is a modular function (if possible, the full resolution and another example of it).

Thanks in advance!

 

[Deleted member 4993]

I've just started studying Integrals and Definite Integrals, and got stuck. I tried applying the Power Rule, but the result doesn't match with the answers.

Here is the question:

. . .\(\displaystyle \mbox{(9) Let }\, f(x)\, =\, \big| \, x^2\, -\, 1\, \big| .\, \mbox{ Then }\, f(0)\, =\, \boxed{\,(1)\,}\, \mbox{ and }\, \)\(\displaystyle \displaystyle \int_0^2\, f(x)\, dx\, =\, \boxed{\, (2)\,}.\)

And the answers are:
1-) 1
2-) 2

How can I solve this?

Thanks in advance.

\(\displaystyle \displaystyle{\int_0^2 (x^2 - 1) dx \ = \ ?}\)

 

[Steven G]

I'd like to know how can I get to the results (more specifically the second question), and what do I have to review/study in order to know how to do so by my own. All I know is about the Power Rule, but doesn't seem to work with that question, considering it is a modular function...


Yes, I know what |x| means. What I want to know is how to get to the 2 result with the Definite Integral calculus, considering it is a modular function (if possible, the full resolution and another example of it).

Thanks in advance!

All you need to know is the power rule. HallsofIvy spelled out what you need to integrate. You say that you know what |x| means. Wonderful, then can you please compute the integral from -3 to 5 of |x|dx. If you can do that then you will get further help.

 

[stapel]

I'd like to know how can I get to the results (more specifically the second question), and what do I have to review/study in order to know how to do so by my own.

Since you've shown no work, we have no way of knowing what all you might need to review (refresh) or study (learn anew). Sorry.

Here is the question:

. . .\(\displaystyle \mbox{(9) Let }\, f(x)\, =\, \big| \, x^2\, -\, 1\, \big| .\, \mbox{ Then }\, f(0)\, =\, \boxed{\,(1)\,}\, \mbox{ and }\, \)\(\displaystyle \displaystyle \int_0^2\, f(x)\, dx\, =\, \boxed{\, (2)\,}.\)

I know what |x| means. What I want to know is how to get to the 2 result with the Definite Integral calculus, considering it is a modular function (if possible, the full resolution and another example of it).

Have you studied integration at all? If so, have you studied any of the "properties" of integrals, such as addition of integrals or splitting one integral apart into two or more, covering different intervals? Have you studied absolute-value functions at all? Have you studied the graphs of these functions? Are you familiar with how to break these apart into sets of linear functions?

Thank you!

 

[Steven G]

Since you've shown no work, we have no way of knowing what all you might need to review (refresh) or study (learn anew). Sorry.



Have you studied integration at all? If so, have you studied any of the "properties" of integrals, such as addition of integrals or splitting one integral apart into two or more, covering different intervals? Have you studied absolute-value functions at all? Have you studied the graphs of these functions? Are you familiar with how to break these apart into sets of linear functions?

Thank you!

I see a theme in your reply- Have you studied? I like that!