Alevel maths q help needed
The curve y=lnx passes through the point (a, b), where a > 1.
The area A is bounded by the x-axis, the line x = a and the curve y=lnx .
The area B is bounded by the x-axis, the y-axis, the line y = b and the curve y=lnx.
The area A is equal in magnitude to the area B.
(a) Show that a satisfies the equation palna+qa+r= 0, where p, q and r are constants to be determined.
I've expressed the area of A but do not know how to express area of B, furthest I got with that was b=lna
The area A is bounded by the x-axis, the line x = a and the curve y=lnx .
The area B is bounded by the x-axis, the y-axis, the line y = b and the curve y=lnx.
The area A is equal in magnitude to the area B.
(a) Show that a satisfies the equation palna+qa+r= 0, where p, q and r are constants to be determined.
I've expressed the area of A but do not know how to express area of B, furthest I got with that was b=lna
https://cdn.discordapp.com/attachments/809953763921362944/1107082788991148093/image.png
This is the answer
(not sure how i can upload the picture directly)
This is the answer
(not sure how i can upload the picture directly)
Original post by xivst
https://cdn.discordapp.com/attachments/809953763921362944/1107082788991148093/image.png
This is the answer
(not sure how i can upload the picture directly)
This is the answer
(not sure how i can upload the picture directly)
You could integrate the inverse function, or note that the required area between the curve and the y axis is the area of the bounding rectangle - A. In the posted working, theyve done the former, though the latter is the simpler way to do it?
(edited 11 months ago)
Original post by xivst
The curve y=lnx passes through the point (a, b), where a > 1.
The area A is bounded by the x-axis, the line x = a and the curve y=lnx .
The area B is bounded by the x-axis, the y-axis, the line y = b and the curve y=lnx.
The area A is equal in magnitude to the area B.
(a) Show that a satisfies the equation palna+qa+r= 0, where p, q and r are constants to be determined.
I've expressed the area of A but do not know how to express area of B, furthest I got with that was b=lna
The area A is bounded by the x-axis, the line x = a and the curve y=lnx .
The area B is bounded by the x-axis, the y-axis, the line y = b and the curve y=lnx.
The area A is equal in magnitude to the area B.
(a) Show that a satisfies the equation palna+qa+r= 0, where p, q and r are constants to be determined.
I've expressed the area of A but do not know how to express area of B, furthest I got with that was b=lna
Could you post an image of the actual question?
Original post by Bruce Wayne59875
Could you post an image of the actual question?
That is exactly how the question is put on the paper but I'll try
Original post by Bruce Wayne59875
Could you post an image of the actual question?
https://cdn.discordapp.com/attachments/809953763921362944/1108036945365438589/image.png
Do you see why its
* the usual integral of the inverse function e^x (reflection in y=x) or
* the area of the bounding rectangle - the integral of ln(x), so aln(a)-A
(edited 11 months ago)
Original post by mqb2766
Do you see why its
* the usual integral of the inverse function e^x (reflection in y=x) or
* the area of the bounding rectangle - the integral of ln(x), so aln(a)-A
* the usual integral of the inverse function e^x (reflection in y=x) or
* the area of the bounding rectangle - the integral of ln(x), so aln(a)-A
I can understand how area A was worked out but I think I'm getting confused with how to visualise area B with the information given in the question
Original post by xivst
I can understand how area A was worked out but I think I'm getting confused with how to visualise area B with the information given in the question
If you sketched A, so the area under ln(x) between 1..a, then put a rectangle around it so with vertices (0,0), (0,ln(a)), (a,0), (a, ln(a)), then B would be the complement of A in that rectangle, so "the area to the left" of ln(x).
Thinking about its shape as a function of y, its "clearly" an exponential as this is the inverse of ln(x) which is the approach used in the solution. However you can also just do
B = area of rectangle - A
So you only really need to work out A (or B).
(edited 11 months ago)
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