Does the integration need to be altered due to the given limits?

[schmiggy]

Homework Statement

Attached scanned image with all variables, given data, and relevant equations.

Homework Equations

See 1.

The Attempt at a Solution

I've also attached my working for the question. Normally these questions are extremely easy, so I feel a little silly for asking, however I've never done one of these where we're given pi as a limit on the x and y planes so I'm not sure I've done this correctly and may have oversimplified it.. I've literally placed an approximate sqrt(pi) marker on my x-axis and twice that on the y-axis for 2sqrt(pi). When I compute the integral the calculations are neat and simple which makes me think it's correct, but I'd appreciate some confirmation on that if possible, thanks :) *EDIT* My answer to question 1 should be 2, not -ve 2.

Also, I'm not really sure what standard procedure is here, but I have a second question which again I think I may have correct but I'm not sure. Rather than hog the forum space with a second thread would I be able to add it to this one? Just in case that's acceptable, I've attached a second image with the question and my attempt at the solution, if not, please disregard the second image :). Regarding the second question, I'm mostly unsure about the limits I've selected.

Images labeled "1" and "2" for questions 1 and 2 respectively.

 

[haruspex]

1) From inspection of the original integral (often a useful check) the answer should be positive. You were ok until the final step.
2) How exactly are you defining theta? The range doesn't look right. Again, the sign is certainly wrong., but I think it's also out by a factor of 2.

 

[schmiggy]

1) From inspection of the original integral (often a useful check) the answer should be positive. You were ok until the final step.
2) How exactly are you defining theta? The range doesn't look right. Again, the sign is certainly wrong., but I think it's also out by a factor of 2.

Thanks for the reply, I amended my original post changing my answer to a positive 2 in bold, but that was most likely while you were formulating your reply, sorry for the confusion - silly error on my part!

As for 2, defining theta is what confused me.. x is less than or equal to 0 which means we're dealing with a semi circle when projected onto the x-y plane. So the +ve y = pi/2, and the negative y = -pi/2 or 3pi/2 doesn't it? When I'm working on limits I usually think of them as start and end points, and with circles I generally go around them anticlockwise.. so I started at pi/2, and it ends at 3pi/2. But I'm guessing that's wrong by your reaction haha.

*edit* I've changed the range to pi/2 to 3pi/2 which yields the same answer, only positive instead of negative, so that's the sign problem fixed at least.. still unsure if that's correct or what I'd need to do to fix it. Alternatively, given it's a semi circle, could I just set the limit from 0 to pi?

 

[SammyS]

...

As for 2, defining theta is what confused me.. x is less than or equal to 0 which means we're dealing with a semi circle when projected onto the x-y plane. So the +ve y = pi/2, and the negative y = -pi/2 or 3pi/2 doesn't it? When I'm working on limits I usually think of them as start and end points, and with circles I generally go around them anticlockwise.. so I started at pi/2, and it ends at 3pi/2. But I'm guessing that's wrong by your reaction haha.

*edit* I've changed the range to pi/2 to 3pi/2 which yields the same answer, only positive instead of negative, so that's the sign problem fixed at least.. still unsure if that's correct or what I'd need to do to fix it. Alternatively, given it's a semi circle, could I just set the limit from 0 to pi?

Having θ go from -π/2 to π/2 would correspond to the region in which x is positive.

Having θ go from π/2 to 3π/2 would correspond to the region in which x is negative, which is what you want.

In the image of your workings, you have θ go from π/2 to -π/2, which will change the sign of the result.

 

[schmiggy]

Thanks SammyS, I've adjusted my ranges, however I still get the result (23pi)/60.. I've attached my working using the amended ranges.

Also, regarding ranges of theta, is it correct then to say when determining the range you work anticlockwise? For example, if it were y is less than or equal to 0 instead, would the range be pi to 2pi?

 

[SammyS]

Thanks SammyS, I've adjusted my ranges, however I still get the result (23pi)/60.. I've attached my working using the amended ranges.

Is that incorrect?

Also, regarding ranges of theta, is it correct then to say when determining the range you work anticlockwise? For example, if it were y is less than or equal to 0 instead, would the range be pi to 2pi?

Yes and yes.

 

[schmiggy]

Perfect, thank you so much, definitely cleared up some confusion of mine!

As for whether it's correct or incorrect, I'm not sure - haruspex said he thought the answer was out by a factor of two.. Going through the working out, it seems like it should be correct. I don't get the answers to these tutorial questions until next week to check.

 

[haruspex]

haruspex said he thought the answer was out by a factor of two..

My mistake, sorry.