# Area Under the arch of y=sinx

Hi guys I'm new to the forum but with year 12 maths coming up next year I assume I'll be coming here a lot.

This question is part of my end of year investigation.

I have scanned the questions and attached them as jpegs.

I was thinking something along the lines of using the triangle to determine the area? Or if the fact that this curve is related to half of a unit circle had something to do with it?

Any help would be much appreciated :)

Thank you, Travis

#### Attachments

SteamKing
Staff Emeritus
Homework Helper
You've got to show an attempt at solving the problem in order to get some help.

CompuChip
Homework Helper
Hi Trav, welcome to the forums. I do hope you'll be coming here - and learning - a lot.
As SteamKing said, please show us your attempt or problem. The questions are leading you in a specific direction. As you've noticed, there is a pretty big hint in the picture in the form of the auxiliary lines that have been drawn, creating for example a triangle. If you show us how far you got with that, we may be able to help you complete it.

Sorry I was unaware that an attempt must be shown, I will be sure to include them in every question from here on out :)
My attempts for each question as follows are:
1. The area of the outer rectangle is pi as pi x 1 = pi
The area for the inner triangle is 1/2bh so 1/2 x pi x 1= 1/2pi or pi/2
this shows that the area under the sine arch must fit the range of pi./2 < x < pi

2.
Triangle area= 1/2 x b x h
1/2 x pi/4 x sin(pi/4)= 0.2776

Trapezium area = (a + b)/2 x h
(sin (pi/4) + sin(pi/2))/2 x pi/4=0.6704

total area = 0.2776x2 + 0.6704x2= 1.896

which is the correct area

I get that sin of (pi/4 or pi/2) is the height at those given points but could someone please give me an explanation as to why this is?

I have found the area for both triangles and trapeziums and multiplied them by 2 to give me the overall area and this is right how ever I cannot make the connection between this and the equation pi/4(1+(root 2))

3. I can use the formula to add up all the areas and I get an answer close to 2 which seems logical as the area under a sine curve is meant to equal 2.
I do not understand how this formula has been derived though?
And for pi to be split into increments of 0.1 I thought the start of the equation should be 1/31.4 as this is pi/0.1?
1/20 would give increments close to 0.15 wouldn't it?