Finding area underneath Log/In graph (1 Viewer)

[cutemouse]

Hi,

I'm pulling hairs with this question. Thanks if you can help me!

Find the exact area bounded by the curve y=In x, the x-axis and the lines x=2 and x=4.

 

[Trebla]

Draw a diagram.

Let the area bounded by the curve y = ln x, the x-axis and the lines x = 2 and x = 4 be A₀

Consider the rectangle bounded by both axes, x = 4 and y = ln 4. Let this be A.
The area of this rectangle is 4ln 4.

Now consider the area bounded by y = ln x and the lines y= ln 2 and y = ln 4 plus the y-axis. Let this be A₁. This is found by:
A₁ = ∫_{ln 2}^{ln 4} e^y dy
= [e^y]_{ln 2}^{ln 4}
= 4 - 2
= 2

Now consider the rectangle bounded by both axes, x = 2 and y = ln 2. Let this be A₂.
The area of this rectangle is 2ln 2.

Now:
A = A₀ + A₁ + A₂
4ln 4 = A₀ + 2 + 2ln2
.: A₀ = 4ln 4 - 2ln 2 - 2
= 8ln2 - 2ln2 - 2
= 6ln 2 - 2 square units

 

[tacogym27101990]

or you could use integration by parts
i loved I.B.P

 

[cutemouse]

Thanks.. I just have 1 question though,

Now consider the area bounded by y = ln x and the lines y= ln 2 and y = ln 4 plus the y-axis. Let this be A₁. This is found by:
A₁ = ∫_{ln 2}^{ln 4} e^y dy
= [e^y]_{ln 2}^{ln 4}
= 4 - 2
= 2

Is this area below or above the graph? If it's above why is that?

Thanks

 

[Trebla]

or you could use integration by parts
i loved I.B.P

Yeah, that would be a lot quicker, but this is Extension 1.

 

[cutemouse]

or you could use integration by parts
i loved I.B.P

Yeah but this is 2U or 3U stuff, so I dunno if I'd get the marks if I used that in a 2U/3U exam

 

[tacogym27101990]

Yeah, that would be a lot quicker, but this is Extension 1.

yeah i know, i just wanted to point it out
i found it fun to do this during the year

 

[lyounamu]

Yeah but this is 2U or 3U stuff, so I dunno if I'd get the marks if I used that in a 2U/3U exam

Yes you can.

You can use any method.

 

[Trebla]

Thanks.. I just have 1 question though,



Is this area below or above the graph? If it's above why is that?

Thanks

Draw y = ln x and the lines y = ln 2 and y = ln 4. Shade the area to the left of the curve, to the right of the y-axis, above y = ln 2 and below y = ln 4.

 

[tacogym27101990]

yeah ive used it in 3 unit assesment tasks to save time

 

[Trebla]

Generally in the external exams for Extension 1, a question like that would have a lead in and use the word "hence" implying that you have to use the area method (or deriving the integral via differentiation).

The questions are usually set so that Extension 2 students don't have an advantage over Extension 1 only students in using techniques exclusive to Extension 2.

 

[cutemouse]

Draw y = ln x and the lines y = ln 2 and y = ln 4. Shade the area to the left of the curve, to the right of the y-axis, above y = ln 2 and below y = ln 4.

Okay yeah, I get that that it's above the graph in this case, but why is that though? Just so I don't get confused with other questions like this.

 

[tacogym27101990]

yeah im pretty sure it did have a lead in question, but the exams were real easy so id just kinda have fun with it.
i once did a simple volume question using shells method, it kept me amused during the test

 

[cutemouse]

yeah ive used it in 3 unit assesment tasks to save time

And I assume you got full marks?

 

[Timothy.Siu]

u can use any method, as long as it doesn't ask for some particular way
but for this u shud change the ln to e

it helps if u draw it.

 

[Trebla]

Okay yeah, I get that that it's above the graph in this case, but why is that though? Just so I don't get confused with other questions like this.

The area the question wants to find is A₀. But A₁ is needed implicity so you can use the rectangles.

 

[tacogym27101990]

And I assume you got full marks?

in the question? yes
but i never got full marks in the exams, always one or two really really stupid mistakes, such as writing a 7 instead of a 9 or such

 

[cutemouse]

u can use any method, as long as it doesn't ask for some particular way
but for this u shud change the ln to e

it helps if u draw it.

Yeah I was asking because I know for loan repayments in 2U, you cannot get full marks if you just use the formula that the General Maths guys get given.

 

[lyounamu]

Yeah I was asking because I know for loan repayments in 2U, you cannot get full marks if you just use the formula that the General Maths guys get given.

No, because there is no formula to it.

It's purely derivation of the formula or something of that sort.

 

[tacogym27101990]

Yeah I was asking because I know for loan repayments in 2U, you cannot get full marks if you just use the formula that the General Maths guys get given.

thats cause the formula given in general doesnt always work in some questions, i can't give a specific example right now but yeah, you have to derive it in 2 unit for this reason.