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Luscinia
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2 Questions here! (I'm not exactly sure if it's allowed, but I want to avoid posting too many threads)
1)Find the following definite integrals by using a trigonometric substitution:
d)1/2∫1dx/(√(2x-x2)
x=asinu
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.
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
Shell:V= 2∏a∫bxhdx
Disc: V=∏a∫br22-r12dx
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!
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!