Question

Evaluate the integral...

Answer 1

How do i solve an integral of an absolute value function?? is there a formula?? i didnt find any.

Answer 2

In general,

If |x| = k

then

x = k or x = -k

for some positive number k

Answer 3

So if y = |x|, then

x = y or x = -y

which can be rearranged to

y = x or y = -x

Answer 4

uh huh

Answer 5

which means that

y = |x-5|

y = x-5 or y = -(x-5)

y = x-5 or y = -x+5

and you get a piecewise function

x-5 if x >= 5
f(x) =
-x+5 if x < 5

Answer 6

so

\[\Large \int_{0}^{10}|x-5|dx\]

breaks up into

\[\Large \int_{0}^{5}(-x+5)dx+\int_{5}^{10}(x-5)dx\]

Answer 7

sorry, im a bit confused about the piecewise function. How do i know the number should be 5 and not, lets say, -5?

Answer 8

also if you graph this you will see that the area under the curve can be found geometrically without calculus....area of 2 triangles

Answer 9

Because the vertex of y = |x-5| is at (5,0). This is where the two pieces effectively switch over.

Answer 10

ohh, i see. but how can in know what is the vertex? is there a formula?

Answer 11

Since y = |x-5| is made up of the pieces of the lines y = x-5 and y = -x+5 (and we're talking about the same y here), this means


y = x-5

-x+5 = x-5 ... Plug in y = -x+5

Solve for x

-x+5 = x-5

5+5 = x+x

10 = 2x

10/2 = x

5 = x

x = 5

So the two lines intersect when x = 5. This is where the junction of the two pieces is. To find the y coordinate, just plug in x = 5 into either equation and evaluate.

Answer 12

In general, the vertex of y = a|x-h|+k is (h,k). In the case of y = |x-5| which is really y = 1|x-5|+0, we can see that h = 5 and k = 0

So the vertex of y = |x-5| is (5,0)

Answer 13

hmmm...i see, great!

Answer 14

I'm glad you do

Answer 15

so, why i put 5 and 0 as the limits of one part and 10 and 0 as the limits of the other part?

Answer 16

how do i get the 5 and 0, and why they are different?

Answer 17

I'm using the idea that

\[\Large \int_{a}^{b}f(x)dx = \int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx\]

where a < c < b

Answer 18

This rule says that you can either find the area under the curve from a to b (left side)

OR

you can find the area under the curve from a to c, then add it to the area under the curve from c to b (right side)


This idea is useful for breaking up areas into smaller and more manageable chunks.

Answer 19

hmmm...but what numbers would be my a,b, and c?

Answer 20

a = 0

b = 10

c = 5

Answer 21

a = 0 and b = 10 are given in the problem

c = 5 is that point where the pieces change

Answer 22

ohhhhh, I see. The value i get for x, right? hmmm

Answer 23

exactly, when you solve that system above

Answer 24

so, then, i find the antiderivatives, and plug in the numbers right...?

Answer 25

Yes, you would then use the idea that

\[\Large \int_{a}^{b} f(x)dx = F(b) - F(a)\]

where F(x) is the antiderivative of f(x)

Answer 26

hmmm...let me try to solve it

Answer 27

ok

Answer 28

is the next step ?