HP15C HP42S integration of trig differs in *rad and *deg

[hansepie]

f(x) = sin(x)

When performing integration in RAD=mode lower limit 0 and upper limit PI/2 it returns 1.0000.

When performing integration in DEG-mode lower limit 0 and upper limit 90 it returns 57.2960.

I expected the same results.

HP-42S and HP-15C give the same results.

What do I miss?

Thank's.

Hans

[Logan]

I don't know for certain, but given that 57.3 is 180/pi, I wonder if it always integrates in radians and then multiplies the result by 57.3 to "convert" the final result into degrees.

[Gene]

Is the result in RAD a measurement in "square radians" and in DEG "square degrees"?

I know that's bad terminology, but I always envision integration as counting the number of squares in units XYZ below a curve from a to b. The size of the square depends on the units on the axes.

?

[Dieter]

(04-27-2018 07:05 PM)hansepie Wrote:  f(x) = sin(x)

When performing integration in RAD=mode lower limit 0 and upper limit PI/2 it returns 1.0000.

That's the correct result: –cos(pi/2) – –cos(0) = 0 – (–1) = 1.

(04-27-2018 07:05 PM)hansepie Wrote:  When performing integration in DEG-mode lower limit 0 and upper limit 90 it returns 57.2960.

That's the correct result as well.

The antiderivative of sin(x) is –cos(x) – but for x in radians!

If you use degrees you actually integrate sin(x° · pi/180°). Here the antiderivative is –180/pi · cos (x°). This leads to a result of 180/pi = 57,2958.

(04-27-2018 07:05 PM)hansepie Wrote:  I expected the same results.

Then you should try GRAD mode as well. This yields 200/pi = 63,662. ;-)

(04-27-2018 07:05 PM)hansepie Wrote:  HP-42S and HP-15C give the same results.

I hope so. ;-)

(04-27-2018 07:05 PM)hansepie Wrote:  What do I miss?

For the mathematical explanation see above.

Here's an idea that may help understand what's going on here: Imagine the graph of sin(x) between 0 and pi/2 or 0 and 90°. The integral is the area between graph an x-axis. The vertical height is the same in both cases: it starts at 0 and finally reaches 1. But the horizontal width is very different: in the one case the area is only 1,57 units wide while in the other case it's a whopping 90 units! So it's clear that the area (i.e. the integral) must be much greater. Actually it is greater by a factor of 90/1,57... = 57,2958.

And finally: welcome to the forum. :-)

Dieter

[hansepie]

(04-27-2018 07:50 PM)Logan Wrote:  I don't know for certain, but given that 57.3 is 180/pi, I wonder if it always integrates in radians and then multiplies the result by 57.3 to "convert" the final result into degrees.

Thanks. Still I find it strange to have to correct the result by multiplying with PI/180 in DEG or with PI/200 in GRA.

I find no reference to this in the manuals.

[zooropa1844]

With those two integrals we are computing different areas so we should't expect the same result. As has been explained in detail by Dieter.

Cheers

[John Cadick]

(04-27-2018 07:05 PM)hansepie Wrote:  f(x) = sin(x)

When performing integration in RAD=mode lower limit 0 and upper limit PI/2 it returns 1.0000.

When performing integration in DEG-mode lower limit 0 and upper limit 90 it returns 57.2960.

I expected the same results.

HP-42S and HP-15C give the same results.

What do I miss?

Thank's.

Hans

You are not getting different answers. You are getting the same answer in different units because 1 radian = 57.2958 degrees. If the upper and lower limits are in radians, so is the answer. If they are in degrees, so is the answer.

As a simple example of this:

A square that is 10m on a side has an area of 100m^2
A square that is 32.8083ft on a side has an area of 1076.391ft^2

The answers are the same because 10m = 32.8083ft and 100m^2 = 1076.391ft^2

John

P.S. For you mathematicians, I know that the equal signs above should probably be a double-ended arrow. Couldn't figure out how to reproduce a double-ended arrow in this posting.

[Dieter]

(04-27-2018 08:16 PM)hansepie Wrote:  Thanks. Still I find it strange to have to correct the result by multiplying with PI/180 in DEG or with PI/200 in GRA.

I find no reference to this in the manuals.

It doesn't multiply a result by anything. The value you got is the true integral.
Please read my first reply in this thread.

Dieter

[hansepie]

Thank you Dieter, your answer is enlightening!

[Dieter]

(04-27-2018 10:46 PM)John Cadick Wrote:  The answers are the same because 10m = 32.8083ft and 100m^2 = 1076.391ft^2
...
P.S. For you mathematicians, I know that the equal signs above should probably be a double-ended arrow.

I am not a mathematician, but I'd say that the equals sign is perfectly OK:

10 m = 10 · 1 m = 10 · 1 m · 3,28 ft/m = 32,8 ft

Quote:Couldn't figure out how to reproduce a double-ended arrow in this posting.

I'd simply add a "<" and ">" to get "<=>". ;-)

For symbols that I use regulary I have prepared a small text file that contains characters like ± ≈ ≠ ≤ ≥ ∞ ← ↑ → ↓ ↔ α β γ Δ Σ etc. On a windows system you may use "charmap" (and a Unicode enabled font) and copy what you want. The character you wanted to use is Unicode 21D4 which is not available in all fonts. You may try Lucida Sans Unicode. If the symbol is available it looks like this: ⇔

Dieter

[hansepie]

(04-27-2018 08:06 PM)Gene Wrote:  Is the result in RAD a measurement in "square radians" and in DEG "square degrees"?

I know that's bad terminology, but I always envision integration as counting the number of squares in units XYZ below a curve from a to b. The size of the square depends on the units on the axes.

?

nope: radians are without units. That was why I was tricked in the first place. See Dieter's reply.

So if you want to regard the curve in respect to units: the y axis has no units and the x axis is in radian (no unit) or in degrees. So no square degrees.

Hans

[zooropa1844]

Cadick said

Quote:If the upper and lower limits are in radians, so is the answer. If they are in degrees, so is the answer.

Only to point out that the units of the result of a definite integral is not in general the units of the limits of integration. It is the case in the example we are dealing with because sin(x) is dimensionless. But in the most general case the units are the units of the integrand times the units of the variable of integration.

Cheers

[John Cadick]

(04-28-2018 01:12 PM)zooropa1844 Wrote:  Cadick said

Quote:If the upper and lower limits are in radians, so is the answer. If they are in degrees, so is the answer.

Only to point out that the units of the result of a definite integral is not in general the units of the limits of integration. It is the case in the example we are dealing with because sin(x) is dimensionless. But in the most general case the units are the units of the integrand times the units of the variable of integration.

Cheers

Thanks for your clarification. You are, of course, absolutely correct. I thought about covering the non-dimensional characteristic of the radian, degree, and various trig functions, but I couldn't come up with a way to do it concisely. Your paragraph explains it very well.

John

[brickviking]

Okay, I've read the above posts, I even understand them—sort of. But I'm a little stuck, and I'll show what I mean (cursor movements are like this).

I entered it like this: EQW INTEGRAL 0 Right PI / 2 Right SIN X Right X Up Up (to highlight the whole equation). Then I hit EVAL and I got PI^2/8 for a result. ->NUM converted that to about 1.2337

I tried doing the same in DEGREES mode:
EQW INTEGRAL 0 [/i]Right[/i] 90 Right SIN X Right X Up Up

When I hit EVAL, it wanted me to switch to Radians mode, and wouldn't eval the sin function otherwise.

My Top line reads DEG XYZ HEX R= 'X'

So, where do I go from here? Have I forgotten something? Apologies for all the hodgepodge sentences, I haven't finished my helmet of tea yet.

(Post 209)

[Carsen]

Hello brickviking.

Since you are in Exact Mode, the 50g is going to try to integrate the equation symbolically. So if you want to get the approximate answer of 57.2957795131, set it to Approx. Mode and DEG Mode. Then EVAL the integral.

Also, the answer is accurate to the number of digits it is FIXED to. So if you want the most precise approx. answer, set it to Std mode.

[brickviking]

(04-29-2018 09:00 AM)Mike (Stgt) Wrote:  

(04-28-2018 10:42 PM)brickviking Wrote:  I entered it like this: EQW INTEGRAL 0 Right PI / 2 Right SIN X Right X Up Up (to highlight the whole equation). Then I hit EVAL and I got PI^2/8 for a result. ->NUM converted that to about 1.2337

Now it's me who is puzzled. Did you enter for PI the letters P and I or did you use the symbol \(\displaystyle \pi\)? ...
Ciao.....Mike

Sorry, Derp moment. I forgot there's mathjax here. I'm too used to forums that don't have the "extras" that this site has, and I had to tinker and prod until I got what I wanted. Let's face it, any time I have to type in \ (\displaystyle \frac {\pi^2}{8} \) just to get \(\displaystyle \frac{\pi^2}{8} \), things get a little hairy. And that's a simple equation, not something all complicated. Still, I should get used to it again, I'll have to face it all when I get back into latex documents.

(Post 210)

[brickviking]

Hm, that's weird. Now I can't even duplicate that result. Sigh. Ah well, I'll let you know what I did when I eventually work it out.

(Post 212)