Corn's Calculus Problems

Alright, so I figured I suck plantain at Calculus, not to mention my teacher gives all of these problems which stretches my imagination past its limit, so here are some problems. I'll bump this thread every once in a while. Some are probably easy for you guys and there's some you've probably never seen before. So...to start it off:

"An ellipse (b^2)(x^2) + (a^2)(y^2) = (a^2)(b^2) with an area of A = pi(a)(b). Assume the area is held constant at 16pi, find the changing rate of b if the a is decreasing at 1/4 units/sec when a = 2 units."

The answer is 1 u/s. How do you get it? I know this answer is 100% correct. I forgot to mention that the answer is sometimes wrong (like....33% of the problems I'll post here will have the wrong answer since we're the first class to receive his own textbook that he made back to front >_>).

Oh, and I forgot to mention that all the problems are word-for-word. My teacher is a Taiwanese who doesn't exactly have the best of grammar skills. O_O

Well, I basically solved the 1st one already, but here's another one.

"Find the area of the region which is enclosed by the curve y^2 = x and the line x + y = 2."

The book's answer is 7/3, but I keep on getting 9/2.

I also got 9/2.

I think your book is wrong.

Hurry for wasting time trying to solve the problem.

let me wolfram that for you
http://www.wolframalpha.com/input/?i=area+between+y^2+%3Dx+and+x%2By%3D2

let me wolfram that for you
http://www.wolframalpha.com/input/?i=area+between+y^2+%3Dx+and+x%2By%3D2

Oh god I never knew Wolfram could do that.

Could it do volume integrals too?

Here is all the Calculus it can do (it can probably do more, I think these are just examples):

http://www.wolframalpha.com/examples/Calculus.html

Not sure if any of them are volume integration, my class hasn't even started integration yet.

Oh god I never knew Wolfram could do that.

Could it do volume integrals too?

Doesn't seem as if it can do volume integrals. It's 2 days since anything was last posted in here, but if you need help with Disc-Washer or Shell method, I could help you with those.

I can understand your problem with grammar. Some Taiwanese people don't exactly have the best grammar. That tends to modify the meaning of things by quite a lot.

Correct, wolfram alpha does not currently support volume integrals.

Another one: I didn't attempt to solve this one, so I'm leaving it up here so when I tackle it later I can see if I get the same answer.

A right conical tank, having a diameter or 10 ft and height of 25 ft, was initially full of water. Then, it is being drained at the rate of pi/1875 (x)^0.5 ft^3/min, where x is the height of the water level.

A) Find a formula for the depth x.
B) the amount of water in the tank at any time t
C) how long will it take for the tank to empty out water?

Anyone? D=. I just got to this problem and I have no idea how to start it.

Another problem.

Can anyone explain to me how to do the derivative y = 9 ^ (-x)? Wolfram is too confusing to follow D=.

Yeah, even though I got a 2350 on the SATs I suck at Calc. Sue me.

The cone one seems annoying.

y = 9^(-x)
y = e^(-x ln 9) <-- exponent rule
y' = e^(-x ln 9) (- ln 9) <-- chain rule
y' = 9^(-x) (- ln 9) <-- put it back
y' = -9^(-x) ln 9.

...yeah, I have no idea how to do the cone one. Tried and I couldn't figure it out.

Russt already solved it, but here's a formula that helps with those type of problems.

Y = A^U, where A is a constant.

Y' = U' * ln(A)* A^U

A bit of a weird one. I think I'm doing something wrong.

"The slope of a curve f(x) is 6x^2(y-5) and the point (0,8) is on the curve. Find the curve."

We're obviously supposed to use Separation of Variables. I did that, and got y = e^(2x^3) + 7. However, the answer is f(x) = 5 + 3e^(x^3).

Is that (6x^2)(y-5) or 6x^(2y-10) ? I think it's the second butttt...?

First one, sorry.

dy/dx=(6x^2)(y-5)

dy/(y-5) = 6 x^2 dx

integrate

ln (y-5) = 2 * x^3 +C

y-5 = e^ (2 * x^3) * e^C

y-5 = c* e^ (2 * x^3)
y = c* e^(2 * x^3) + 5

(0,8) is on the curve so...

8 = c * e^0 + 5
c = 3
^ lols
and thus

y = 3 e^(2 * x^3) +5.

So...I got another problem.

Integral (e^x*csc((e^x)+1)) dx

I'm pretty sure there's a simpler answer than Wolfram's answer. I'm not entirely sure if Integration by Parts will cut it.