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Finding the area enclosed by the graphs of: $y=xleft(x-4right)^2$ ,$y=4x-x^2$.
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Find the area enclosed by the graphs of:
$y=xleft(x-4right)^2$
, $y=4x-x^2$.
My answer was $frac712+frac454$, but apparently this is wrong: the right answer is $frac372$ according to the textbook.
Intersections occur where $x(x - 4)^2 = x(4 - x)$.
So $x(x - 4)(x - 4 + 1) = 0$ which has solutions at $x = 0$, $x = 3$ and $x = 4$.
The difference equation is $y = x(x - 4)^2 - (4x + x^2) = x^3 - 9x^2 + 12x$.
beginalign textEnclosed area & = int_0^4 lvert x^3 - 9x^2 + 12x rvert , dx \
& = int_0^3 x^3 - 9x^2 + 12x , dx + int_3^4 -x^3 + 9x^2 - 12x , dx \
& = left[ frac14 x^4 - 3x^3 + 6x^2 right]_0^3 + left[ -frac14 x^4 + 3x^3 - 6x^2 right]_3^4 \
& = left( frac814 - 81 + 54 right) - left( 0 right) + left( -64 + 192 - 96 right) - left( -frac814 + 81 - 54 right) \
& = -frac274 - 0 + 32 - frac274 \
& = frac372 endalign
It appears that they changed the $-x^2$ to $x^2$ in the latter equation when finding the difference equation. Why did they do that?
integration proof-verification definite-integrals area
edited Aug 3 at 6:14
Bladewood
19513
asked Aug 3 at 3:30
Cheks Nweze
486
 |Â
show 1 more comment
up vote
3
down vote
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Find the area enclosed by the graphs of:
$y=xleft(x-4right)^2$
, $y=4x-x^2$.
My answer was $frac712+frac454$, but apparently this is wrong: the right answer is $frac372$ according to the textbook.
Intersections occur where $x(x - 4)^2 = x(4 - x)$.
So $x(x - 4)(x - 4 + 1) = 0$ which has solutions at $x = 0$, $x = 3$ and $x = 4$.
The difference equation is $y = x(x - 4)^2 - (4x + x^2) = x^3 - 9x^2 + 12x$.
beginalign textEnclosed area & = int_0^4 lvert x^3 - 9x^2 + 12x rvert , dx \
& = int_0^3 x^3 - 9x^2 + 12x , dx + int_3^4 -x^3 + 9x^2 - 12x , dx \
& = left[ frac14 x^4 - 3x^3 + 6x^2 right]_0^3 + left[ -frac14 x^4 + 3x^3 - 6x^2 right]_3^4 \
& = left( frac814 - 81 + 54 right) - left( 0 right) + left( -64 + 192 - 96 right) - left( -frac814 + 81 - 54 right) \
& = -frac274 - 0 + 32 - frac274 \
& = frac372 endalign
It appears that they changed the $-x^2$ to $x^2$ in the latter equation when finding the difference equation. Why did they do that?
integration proof-verification definite-integrals area
edited Aug 3 at 6:14
Bladewood
19513
asked Aug 3 at 3:30
Cheks Nweze
486
I'm not sure what you're referring to with the change of sign... which line exactly?
– spaceisdarkgreen
Aug 3 at 4:03
@spaceisdarkgreen hi, thanks for replying. The line that goes “The difference equation is...†(line 3 of the image)
– Cheks Nweze
Aug 3 at 4:05
7
You are correct, that is a typo. And furthermore it propagates all the way to the end. Your answer is correct
– spaceisdarkgreen
Aug 3 at 4:08
Awesome. Cheers!
– Cheks Nweze
Aug 3 at 4:09
Just a detail $x^3-colorred7 x^2+12 x$ would be better (if we refer to the title).
– Claude Leibovici
Aug 3 at 11:58
 |Â
show 1 more comment
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Find the area enclosed by the graphs of:
$y=xleft(x-4right)^2$
, $y=4x-x^2$.
My answer was $frac712+frac454$, but apparently this is wrong: the right answer is $frac372$ according to the textbook.
Intersections occur where $x(x - 4)^2 = x(4 - x)$.
So $x(x - 4)(x - 4 + 1) = 0$ which has solutions at $x = 0$, $x = 3$ and $x = 4$.
The difference equation is $y = x(x - 4)^2 - (4x + x^2) = x^3 - 9x^2 + 12x$.
beginalign textEnclosed area & = int_0^4 lvert x^3 - 9x^2 + 12x rvert , dx \
& = int_0^3 x^3 - 9x^2 + 12x , dx + int_3^4 -x^3 + 9x^2 - 12x , dx \
& = left[ frac14 x^4 - 3x^3 + 6x^2 right]_0^3 + left[ -frac14 x^4 + 3x^3 - 6x^2 right]_3^4 \
& = left( frac814 - 81 + 54 right) - left( 0 right) + left( -64 + 192 - 96 right) - left( -frac814 + 81 - 54 right) \
& = -frac274 - 0 + 32 - frac274 \
& = frac372 endalign
It appears that they changed the $-x^2$ to $x^2$ in the latter equation when finding the difference equation. Why did they do that?
integration proof-verification definite-integrals area
edited Aug 3 at 6:14
Bladewood
19513
asked Aug 3 at 3:30
Cheks Nweze
486
Find the area enclosed by the graphs of:
$y=xleft(x-4right)^2$
, $y=4x-x^2$.
My answer was $frac712+frac454$, but apparently this is wrong: the right answer is $frac372$ according to the textbook.
Intersections occur where $x(x - 4)^2 = x(4 - x)$.
So $x(x - 4)(x - 4 + 1) = 0$ which has solutions at $x = 0$, $x = 3$ and $x = 4$.
The difference equation is $y = x(x - 4)^2 - (4x + x^2) = x^3 - 9x^2 + 12x$.
beginalign textEnclosed area & = int_0^4 lvert x^3 - 9x^2 + 12x rvert , dx \
& = int_0^3 x^3 - 9x^2 + 12x , dx + int_3^4 -x^3 + 9x^2 - 12x , dx \
& = left[ frac14 x^4 - 3x^3 + 6x^2 right]_0^3 + left[ -frac14 x^4 + 3x^3 - 6x^2 right]_3^4 \
& = left( frac814 - 81 + 54 right) - left( 0 right) + left( -64 + 192 - 96 right) - left( -frac814 + 81 - 54 right) \
& = -frac274 - 0 + 32 - frac274 \
& = frac372 endalign
It appears that they changed the $-x^2$ to $x^2$ in the latter equation when finding the difference equation. Why did they do that?
integration proof-verification definite-integrals area
edited Aug 3 at 6:14
Bladewood
19513
asked Aug 3 at 3:30
Cheks Nweze
486
edited Aug 3 at 6:14
Bladewood
19513
edited Aug 3 at 6:14
Bladewood
19513
edited Aug 3 at 6:14
Bladewood
19513
19513
asked Aug 3 at 3:30
Cheks Nweze
486
asked Aug 3 at 3:30
Cheks Nweze
486
asked Aug 3 at 3:30
Cheks Nweze
486
486
I'm not sure what you're referring to with the change of sign... which line exactly?
– spaceisdarkgreen
Aug 3 at 4:03
@spaceisdarkgreen hi, thanks for replying. The line that goes “The difference equation is...†(line 3 of the image)
– Cheks Nweze
Aug 3 at 4:05
7
You are correct, that is a typo. And furthermore it propagates all the way to the end. Your answer is correct
– spaceisdarkgreen
Aug 3 at 4:08
Awesome. Cheers!
– Cheks Nweze
Aug 3 at 4:09
Just a detail $x^3-colorred7 x^2+12 x$ would be better (if we refer to the title).
– Claude Leibovici
Aug 3 at 11:58
 |Â
show 1 more comment
I'm not sure what you're referring to with the change of sign... which line exactly?
– spaceisdarkgreen
Aug 3 at 4:03
@spaceisdarkgreen hi, thanks for replying. The line that goes “The difference equation is...†(line 3 of the image)
– Cheks Nweze
Aug 3 at 4:05
7
You are correct, that is a typo. And furthermore it propagates all the way to the end. Your answer is correct
– spaceisdarkgreen
Aug 3 at 4:08
Awesome. Cheers!
– Cheks Nweze
Aug 3 at 4:09
Just a detail $x^3-colorred7 x^2+12 x$ would be better (if we refer to the title).
– Claude Leibovici
Aug 3 at 11:58
I'm not sure what you're referring to with the change of sign... which line exactly?
– spaceisdarkgreen
Aug 3 at 4:03
I'm not sure what you're referring to with the change of sign... which line exactly?
– spaceisdarkgreen
Aug 3 at 4:03
@spaceisdarkgreen hi, thanks for replying. The line that goes “The difference equation is...†(line 3 of the image)
– Cheks Nweze
Aug 3 at 4:05
@spaceisdarkgreen hi, thanks for replying. The line that goes “The difference equation is...†(line 3 of the image)
– Cheks Nweze
Aug 3 at 4:05
7
7
You are correct, that is a typo. And furthermore it propagates all the way to the end. Your answer is correct
– spaceisdarkgreen
Aug 3 at 4:08
You are correct, that is a typo. And furthermore it propagates all the way to the end. Your answer is correct
– spaceisdarkgreen
Aug 3 at 4:08
Awesome. Cheers!
– Cheks Nweze
Aug 3 at 4:09
Awesome. Cheers!
– Cheks Nweze
Aug 3 at 4:09
Just a detail $x^3-colorred7 x^2+12 x$ would be better (if we refer to the title).
– Claude Leibovici
Aug 3 at 11:58
Just a detail $x^3-colorred7 x^2+12 x$ would be better (if we refer to the title).
– Claude Leibovici
Aug 3 at 11:58
 |Â
show 1 more comment
1 Answer
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You are correct. That is a typo. Furthermore it propagates all the way to the end. Your answer is correct.
answered Aug 3 at 18:36
spaceisdarkgreen
27.1k21546
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
You are correct. That is a typo. Furthermore it propagates all the way to the end. Your answer is correct.
answered Aug 3 at 18:36
spaceisdarkgreen
27.1k21546
add a comment |Â
up vote
0
down vote
accepted
You are correct. That is a typo. Furthermore it propagates all the way to the end. Your answer is correct.
answered Aug 3 at 18:36
spaceisdarkgreen
27.1k21546
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
You are correct. That is a typo. Furthermore it propagates all the way to the end. Your answer is correct.
answered Aug 3 at 18:36
spaceisdarkgreen
27.1k21546
You are correct. That is a typo. Furthermore it propagates all the way to the end. Your answer is correct.
answered Aug 3 at 18:36
spaceisdarkgreen
27.1k21546
answered Aug 3 at 18:36
spaceisdarkgreen
27.1k21546
answered Aug 3 at 18:36
spaceisdarkgreen
27.1k21546
answered Aug 3 at 18:36
spaceisdarkgreen
27.1k21546
27.1k21546
add a comment |Â
add a comment |Â
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I'm not sure what you're referring to with the change of sign... which line exactly?
– spaceisdarkgreen
Aug 3 at 4:03
@spaceisdarkgreen hi, thanks for replying. The line that goes “The difference equation is...†(line 3 of the image)
– Cheks Nweze
Aug 3 at 4:05
7
You are correct, that is a typo. And furthermore it propagates all the way to the end. Your answer is correct
– spaceisdarkgreen
Aug 3 at 4:08
Awesome. Cheers!
– Cheks Nweze
Aug 3 at 4:09
Just a detail $x^3-colorred7 x^2+12 x$ would be better (if we refer to the title).
– Claude Leibovici
Aug 3 at 11:58