Integration q (1 Viewer)

[kkk579]

confused on why its not f(x) -x for the 2nd integral and x - f(x) for 1st integral for 14a

 

[cossine]

confused on why its not f(x) -x for the 2nd integral and x - f(x) for 1st integral for 14aView attachment 47471View attachment 47472

I can't understand your question. Make sure to provide more context

 

[wizzkids]

confused on why its not f(x) -x for the 2nd integral and x - f(x) for 1st integral for 14a

If you do it your way you will get a negative value for the area.
The suggested solution relies on the fact the function and its inverse are reflections about the line y=x.
So draw a line on the graph for y=x.
Using symmetry, you can see that the areas above and below the line y=x are equal areas.
For the interval to you need to integrate and for the interval to you need to integrate so that the areas both come out positive.

 

[kkk579]

If you do it your way you will get a negative value for the area.
The suggested solution relies on the fact the function and its inverse are reflections about the line y=x.
So draw a line on the graph for y=x.
Using symmetry, you can see that the areas above and below the line y=x are equal areas.
For the interval to you need to integrate and for the interval to you need to integrate so that the areas both come out positive.

could you put an absolute on outside? cos i was taught that when calculating volume which is generated by rotating it around y axis its always the curve you hit first when u draw a line from the right to left - the other curve

 

[Eagle Mum]

cos i was taught that when calculating volume which is generated by rotating it around y axis its always the curve you hit first when u draw a line from the right to left - the other curve

This question has nothing to do with volumes of shapes created by rotation around the y-axis so the specific ‘tip’ you were taught doesn’t apply to this question. It’s usually not a good idea to remember specific ‘tips‘ without understanding the underlying principles (which often then negates the need for the tip(s)) because of the risk of applying them incorrectly.

It’s inappropriate to ‘put an absolute outside’, because one cannot just add an operation which only produces the right solution in specific case(s).

Wizzkids explanation nails it.

 

[kkk579]

This question has nothing to do with volumes of shapes created by rotation around the y-axis so the specific ‘tip’ you were taught doesn’t apply to this question. It’s usually not a good idea to remember specific ‘tips‘ without understanding the underlying principles (which often then negates the need for the tip(s)) because of the risk of applying them incorrectly.

It’s inappropriate to ‘put an absolute outside’, because one cannot just add an operation which only produces the right solution in specific case(s).

Wizzkids explanation nails it.

Oh wait sorry i wrote it wrong but besides the rotating volume part the rest still applies from what i was taught

 

[kkk579]

This question has nothing to do with volumes of shapes created by rotation around the y-axis so the specific ‘tip’ you were taught doesn’t apply to this question. It’s usually not a good idea to remember specific ‘tips‘ without understanding the underlying principles (which often then negates the need for the tip(s)) because of the risk of applying them incorrectly.

It’s inappropriate to ‘put an absolute outside’, because one cannot just add an operation which only produces the right solution in specific case(s).

Wizzkids explanation nails it.

How would you know before doing the q that integrating the expressions in that order will result in a positive result?

 

[wizzkids]

Q: How would you know before doing the q that integrating the expressions in that order will result in a positive result?

A: Look at the graphs of the functions. Identify the areas that lie below each function. In each interval you have to subtract the smaller area from the larger area in order for the difference to come out positive.

 

[cheesynooby]

I may be completely misunderstanding the situation but maybe you didn't realise that the dotted line is not f(x) but its inverse? I initially thought the dotted line was f(x) because of the way the graph is drawn, in which case what you said would be correct.

 

[kkk579]

I may be completely misunderstanding the situation but maybe you didn't realise that the dotted line is not f(x) but its inverse? I initially thought the dotted line was f(x) because of the way the graph is drawn, in which case what you said would be correct.

No i recognise that but because of the rule that i was taught which is outer graph - inner graph is why i got the functions the opposite way of whats written in the solutions

 

[Eagle Mum]

No i recognise that but because of the rule that i was taught which is outer graph - inner graph is why i got the functions the opposite way of whats written in the solutions

could you put an absolute on outside? cos i was taught that when calculating volume which is generated by rotating it around y axis its always the curve you hit first when u draw a line from the right to left - the other curve

The reason that you go from right to left when rotating around the y axis is because the x values on the right are larger (more positive) on the right (for positive values of x), so that is the outer and larger radius.

Similarly, for questions pertaining to areas under the curve (where the x axis is the frame of reference), the curve with the larger y value is usually the minuend. It's best to look at the sketched graph for each specific question then to memorise tips without understanding when to apply them (not trying to be critical, just encouraging deeper understanding for better performance).