Question

\[\int_{-3}^{3} |x+1|dx\]
Please, help

Answer 1

Integrate separately on parts of the domain that are convenient is the quick tip. :)

Answer 2

Show me, please.

Answer 3

Or, just draw it and find the area. :)

Answer 4

Will do, I'll do the drawing method since it's more clever.

Answer 5

I show you my confusion.

Answer 6

no confuse, hehehe.. I am sorry. I posted the wrong question. But I would like to know how to solve it traditionally

Answer 7

No worries. :) You should be able to see from the graph that the answer is the area of two triangles. But I will do the traditional (algebraic nastiness) approach as well.

Answer 8

I meant integrate separately..........

Answer 9

First, we need the definition of what |x + 1| means:

|x + 1| = { x + 1, for x >= -1; -(x + 1), for x < -1.

This is the usual, "make the number positive" rule we remember in our heads expressed algebraically.

Answer 10

It may take a bit of pondering to fully see that. :)

Answer 11

Then, this naturally gives us two convenient domains. (-infinity, -1) and (-1, +infinity). The split point is at x = -1.

Answer 12

We want (-3, 3) so we break this into the domains (-3, -1), and (-1, 3).

Answer 13

Then we integrate using the definition of absolute value.

Answer 14

\[\int\limits_{-3}^3 |x + 1| \ dx = \int\limits_{-3}^{-1} -(x+1) \ dx + \int\limits_{-1}^3 x+1 \ dx\]

Answer 15

The integration can then be carried out as usual.

Answer 16

Got you. Thanks a lot

Answer 17

No problems! :)