Calculus AB AP exam question from 2010 (Area/Integration)

[Kahing]

Hello, I am currently studying previous AP Calculus AB exams and I was wondering if someone could explain how to solve the following as clearly as possible so I have a better understanding of these problems in hope to prepare my for the exam next month.

R is the shaded region in the Figure in the first quadrant bounded by the graph of y = 4ln(3-x), the horizontal line is y = 6, and the vertical line is x = 2

a) Find the Area of R
*I have tried using the formula A = integration of a to b, [f(x) - g(x)] dx
but my answer does not come out to be 6.816 or 6.817

b) Find the volume of the solid generated when R is revolved around the horizontal line y = 8
*I have tried the Disk and Washer Methods, but still no luck, I may possibly be doing it wrong since my teacher has not explained this chapter to my class yet. The answer for b) is 26.266

c) The Region R is the base of a solid. For this solid, each cross section perpendicular to the X-axis is a square, find the volume of the solid.


The source below has a picture and official pdf file from collegeboard.org if you need a visual/ clearer explanation.

Source: [COLLEGEBOARD] http://apcentral.collegeboard.com/apc/pu…

*I Am not asking for anyone to do my homework, passing the AP exam is self motivation, and please, do not post anything unless it is helpful. Being in Calculus, I'd assume the students are not lazy to do the homework by themselves since math is built upon foundations upon foundations, its just a matter of understanding. Thank you everyone.

 

[Nebuchadnezza]

this is not the correct forum for asking questions related to subjects. Will ask for the thread to be moved. Though I will give you the solution to your problem

1. ALWAYS draw your problems, if possible.

Here we can see that the area we are looking for are sort of the shape from A following the curve to C then to D then to E then to F and back to A. We can split this area into two parts.

2. Find the intersectionpoint between your function and the y-axis (Find A)
3. Find the brown area, integration might be smart here.
4. Find the area of the rectangle ABCD
6. Find the area of ADEF
7. The area you are looking for is ABCD - j + ADEF

 

[Kahing]

I have followed the steps you have given me, but i still do not come up with the same solution as the answer key :/

Find the Y intercept for y = 4ln(3-x) = y = 4.3944
Find the Area under the function y = 4ln(3-x)
Find the area of the rectangle ABCD = ABCD = 8.7889
Find the area of ADEF= ADEF = 3.2111
The area you are looking for is ABCD - j + ADEF

j = f(2) - f(0)

 

[hunt_mat]

For your integral you need to know the two values of x to integrate to and from, you know one, that is x=2, to find the other one:

$$6=4\log (3-x)\Rightarrow e^{\frac{3}{2}}=3-x\Rightarrow x=3-e^{\frac{3}{2}}=-1.4817$$

Then all you have to do is integrate.

 

[Kahing]

For your integral you need to know the two values of x to integrate to and from, you know one, that is x=2, to find the other one:

$$6=4\log (3-x)\Rightarrow e^{\frac{3}{2}}=3-x\Rightarrow x=3-e^{\frac{3}{2}}=-1.4817$$

Then all you have to do is integrate.

how come the other value is not 0 when it is the y intercept?

please show a step by step explanation, thank you

 

[hunt_mat]

Because it isn't the y-intercept, it is at y=6, you have to ask where the liny y=6 intersects the graph.

 

[vela]

Because it isn't the y-intercept, it is at y=6, you have to ask where the liny y=6 intersects the graph.

You missed the bit in the original problem that says R is in the first quadrant so it ends up bounded by x=0 on the left.

 

[hunt_mat]

Oh right. That would explain the confusion then.

 

[vela]

a) Find the Area of R
*I have tried using the formula A = integration of a to b, [f(x) - g(x)] dx
but my answer does not come out to be 6.816 or 6.817

Show us your work in detail. You have the right idea, but you're apparently messing up during the execution.