- #1

Luscinia

- 17

- 0

## Homework Statement

1)Find the following definite integrals by using a trigonometric substitution:

d)

_{1/2}∫

^{1}dx/(√(2x-x

^{2})

## Homework Equations

x=asinu

## The Attempt at a Solution

From a previous question, I found that the indefinite integral of ∫dx/(√(2x-x

^{2}) 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=x

^{2}about (a) the line x=2 (b) the line y=4

## Homework Equations

Shell:V= 2∏

_{a}∫

^{b}xhdx

Disc: V=∏

_{a}∫

^{b}r

_{2}

^{2}-r

_{1}

^{2}dx

## The Attempt at a Solution

**a)**

x+2=x

^{2}

0=x

^{2}-x+2

(-b±√(b

^{2}-4ac))/2a=(1±√(1-4(1)(-2))/2(1)=(1±3)/2

x=-1 and x=2

2∏

_{-1}∫

^{2}x*(x+2-x

^{2})dx

2∏(1/3*x

^{3}+x

^{2}-1/4x

^{4}|

_{-1}

^{2}

2∏(8/3-5/12)=

9∏/2

The correct answer was supposed to be 27∏/2

**b)**

∏

_{a}∫

^{b}(4-(x

^{2})

^{2}(4-(x+2))

^{2}dx

∏

_{a}∫

^{b}12-9x

^{2}+4x-x

^{4}dx

∏(12x-3x

^{3}+2x[sup2]-1/5x[sup5])|

_{-1}

^{2}

∏[(24-24+8-32/5)-(-12+3+2+1/5)]=

42∏/5

The answer is supposed to be 108∏/5

Thanks!