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C4 Integration question

Would appreciate some help with this question I'm finding tricky.

By using the substitution given, show that the LHS gives the RHS [6]

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Reply 1

Clarity Incognito

Original post by latslikeabat

Would appreciate some help with this question I'm finding tricky.

By using the substitution given, show that the LHS gives the RHS [6]

Whenever you post, explain what you have attempted to do or better, write that what you have done so far, where you have got to so then someone can better give you hints.

Reply 2

latslikeabat

oops my bad.

well i got:

$\frac{dx}{da}$

I then used the substition to form

$\int$

and hence

$\int$

=

$\int$

=

$\int$

=

$\int$

I wasn't sure what to do after this but considered using parts, however that seemed overly complex.

(edited 13 years ago)

Reply 3

Clarity Incognito

Original post by latslikeabat

Ok great, can work with that. You've got the right idea but the notation is a bit sloppy though, in terms of what you're integrating with respect to. After the substitution, you want to be integrating with respect to theta, not a, a is just a constant here.

\displaystyle \int^{\frac{a}{2}}_a \sqrt{a^2-x^2} \ dx

x = a \sin \theta

\dfrac{dx}{d \theta} = a \cos \theta

dx = a \cos \theta d \theta

Changing limits and subbing in to put in terms of theta.

\displaystyle \int ^{\frac{\pi}{6}}_{\frac{\pi}{2}} \sqrt{a^2cos^2\theta}acos \theta \ d \theta

Reply 4

latslikeabat

Original post by Clarity Incognito

\displaystyle \int^{\frac{a}{2}}_a \sqrt{a^2-x^2} \ dx

x = a \sin \theta

\dfrac{dx}{d \theta} = a \cos \theta

dx = a \cos \theta d \theta

Changing limits and subbing in to put in terms of theta.

\displaystyle \int ^{\frac{\pi}{6}}_{\frac{\pi}{2}} \sqrt{a^2cos^2\theta}acos \theta \ d \theta

Ah ok , that makes a lot more sense.

and so after that would you get:

$\int$

and then use the double angle formula so that:

$\cos$

After that point would you use parts so that:

$u$

and

$dv$

?

Also what is 'u' differentiated w.r.t?

Reply 5

irfan91

this is how you do it

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Reply 6

tiny hobbit

Original post by latslikeabat

Since a is a constant you don't need to use "by parts". What do you get if you integrate 7 cos x? Just 7 sin x. Treat the a^2 like a 7.

Reply 7

Clarity Incognito

Original post by latslikeabat

Ah ok , that makes a lot more sense.
Also what is 'u' differentiated w.r.t?

a is just a constant so a^2 is also a constant. You can take it out of the integral. You're integrating with respect to theta.

Original post by irfan91

this is how you do it

Just a note, do you post full solutions, the point of the forum is to help users gain a better understanding, not to give them solutions. Please read the forum rules.

Reply 8

irfan91

Original post by Clarity Incognito

a is just a constant so a^2 is also a constant. You can take it out of the integral. You're integrating with respect to theta.

Just a note, do you post full solutions, the point of the forum is to help users gain a better understanding, not to give them solutions. Please read the forum rules.

lool sorry officer

Reply 9

Clarity Incognito

Original post by irfan91

lool sorry officer

Haha, didn't mean for it to sound so angry, I just wanted to point it out, it's quite usual for members to let newer members know. :h:

EDIT: Also, I just looked through your working and you don't account for the limits of substitution properly. The top limit should be pi/6 throughout and the bottom pi/2 unless you swap them and put a minus out front. The integral actually gives area under the x axis and as area is positive, you take the magnitude of the answer.

(edited 13 years ago)

Reply 10

latslikeabat

Ok, now I get it.

Thanks for the help.

Reply 11

irfan91

Original post by Clarity Incognito

Haha, didn't mean for it to sound so angry, I just wanted to point it out, it's quite usual for members to let newer members know. :h:

its all good, yeah i tried it first time with the limits the other way round and i didn't get the RHS so i just flipped it to get the RHS, thanks for the advice and forum rules