Integration help. How come this doesnt work?

Okay so there are 2 lines
y=7-x^2
&
y=x+5
(http://graphsketch.com/)
I found the intercept point of x=1 and x=-2
I then worked out the co-ordinates to be (1,3) (-2,6)
I then worked out the area of the but under the y= x+5 curve, so when I integrate the y = -x^2 +7 I could just - this area from it.
So i did 6 (height) * 3 (base) to equal 18, then I - 4.5 as it was half of 9 (the area of the aditional bit we dont need.
So the area is 13.5 underneath the line.
I then integrate y= -x^2 +7 to make:
3x^3/3 + 7x
I then factor in the 1 to get:
1/3 + 7
so it is 7 1/3 - (-2^3 /3 + 7*-2)
7 1/3 -( -8/3 + -14)
7.333 -(-16.666)
7.333+16.666=24
I then did 24 - my earlier calculated area under it (13.5) to get 10.5.
However the answer is 4.5 and I cannot see where I went wrong. I understand you are supposed to integrate bot lines togetherness, but why does this not work?
Integrating the y= -x^2 +7 gives us the area of everything underneath it, to the x axis from 1 to -2. By minusing the area under it, (13.5) I cannot understand how, logically it doesnt work? Anyone care to weigh in, would really help.
Currently teaching myself integration so take it easy & explain clearly please. Only started a hour ago xD

Reply 1

SH0405

15

Original post by Danny.L

Okay so there are 2 lines
y=7-x^2
&
y=x+5
(http://graphsketch.com/)
I found the intercept point of x=1 and x=-2
I then worked out the co-ordinates to be (1,3) (-2,6)
I then worked out the area of the but under the y= x+5 curve, so when I integrate the y = -x^2 +7 I could just - this area from it.
So i did 6 (height) * 3 (base) to equal 18, then I - 4.5 as it was half of 9 (the area of the aditional bit we dont need.
So the area is 13.5 underneath the line.
I then integrate y= -x^2 +7 to make:
3x^3/3 + 7x
I then factor in the 1 to get:
1/3 + 7
so it is 7 1/3 - (-2^3 /3 + 7*-2)
7 1/3 -( -8/3 + -14)
7.333 -(-16.666)
7.333+16.666=24
I then did 24 - my earlier calculated area under it (13.5) to get 10.5.
However the answer is 4.5 and I cannot see where I went wrong. I understand you are supposed to integrate bot lines togetherness, but why does this not work?
Integrating the y= -x^2 +7 gives us the area of everything underneath it, to the x axis from 1 to -2. By minusing the area under it, (13.5) I cannot understand how, logically it doesnt work? Anyone care to weigh in, would really help.
Currently teaching myself integration so take it easy & explain clearly please. Only started a hour ago xD

The integral of 7-x² with respect to x is 7x - ⅓x³; not what you have put. From there, it should all work out.

Hope that helps.

Reply 2

Danny.L

OP

3

Original post by SH0405

The integral of 7-x² with respect to x is 7x - ⅓x³; not what you have put. From there, it should all work out.

Hope that helps.

Thank you

Teaching yourself is hard D:
Currently struggling with the trapezium rule in intergeneration! I dont understand the formula! :angry:

Reply 3

SH0405

15

Original post by Danny.L

Thank you

Teaching yourself is hard D:
Currently struggling with the trapezium rule in intergeneration! I dont understand the formula! :angry:

It's easier to visualise and understand if you imagine turning the graph on its side, so the trapezia look more 'normal'. The height then actually becomes the height, which is the interval on the axis, and is obviously the same for each trapezium. The parallel sides of each trapezium are the y values that are obtained when each x value is put into the given function.

The formula simply sums the area of each trapezium. ½h can be factored out, and all but the end two y values are used twice, which explains the 2 in the formula. It's pretty difficult to explain, so this video may clear things up for you: https://www.youtube.com/watch?v=wRtnU_T0VnU. (He sounds like a leprechaun, but a helpful one.)