Trig Substitution and Volume of Solids

[Luscinia]

2 Questions here! (I'm not exactly sure if it's allowed, but I want to avoid posting too many threads)

Homework Statement

1)Find the following definite integrals by using a trigonometric substitution:
d)_1/2∫¹dx/(√(2x-x²)

Homework Equations

x=asinu

The Attempt at a Solution

From a previous question, I found that the indefinite integral of ∫dx/(√(2x-x²) was arcsin(x-1) + C
Using FTC part 2, I got:
arcsin(1-1)-arcsin(1/2-1)=arcsin0-arcsin(-1/2)=0- -∏/6=∏/6

The answer the teacher gave me was ∏/2
I'm have no idea where I have made a mistake.

Homework Statement

Find the volumes of the solids generated by revolving the region bounded by the curves y=x+2 and y=x² about (a) the line x=2 (b) the line y=4

Homework Equations

Shell:V= 2∏_a∫^bxhdx
Disc: V=∏_a∫^br₂²-r₁²dx

The Attempt at a Solution

a)
x+2=x²
0=x²-x+2
(-b±√(b²-4ac))/2a=(1±√(1-4(1)(-2))/2(1)=(1±3)/2
x=-1 and x=2

2∏_-1∫²x*(x+2-x²)dx
2∏(1/3*x³+x²-1/4x⁴|_-1²
2∏(8/3-5/12)=

9∏/2
The correct answer was supposed to be 27∏/2


b)
∏_a∫^b(4-(x²)²(4-(x+2))²dx
∏_a∫^b12-9x²+4x-x⁴dx
∏(12x-3x³+2x[sup2]-1/5x[sup5])|_-1²
∏[(24-24+8-32/5)-(-12+3+2+1/5)]=
42∏/5
The answer is supposed to be 108∏/5


Thanks!

 

[sandy.bridge]

For d) It is not
arcsin(x-1)

 

[I like Serena]

Welcome to PF, Luscinia!

No mistake, your answer to (d) is correct.
It is arcsin(x-1) + C and it evaluates to ∏/6.
(Sorry sandy.bridge.)

 

[sandy.bridge]

Oops, you're right. I presume that your issue was with regards to your lower and upper limits. Did you remember to alter them during substitution?

 

[Luscinia]

Thanks for the quick answer!
I'm not sure why I need to replace the upper and lower limit since the final integral for ∫dx/(√(2x-x²) uses the variable x as well.
(Though I honestly forgot about them and I'm not too sure if I know how to do it properly)
I had used (x-1)=sinu as substitution when I had integrated so arcsin(x-1)=arcsin(sinu)=u

(1/2-1)=sinu
arcsin(-1/2)=u=-∏/6 (lower limit)

(1-1)sinu
arcsin(0)=u=0 (upper limit)

using FTC:
0--∏/6=∏/6?

Maybe my teacher made a mistake?

 

[sandy.bridge]

Okay, I just did the question and got the same answer as you. Your teacher must have given you a false answer.