# Math Help - Avg. Value

1. ## Avg. Value

$\int_{pi/4}^{pi/2} sinx dx$
I tried finding the average value of the function and came out with the answer .5554 sq. units. Is this correct?

$\int_{pi/4}^{pi/2} sinx dx$
I tried finding the average value of the function and came out with the answer .5554 sq. units. Is this correct?
No, I'm sorry

I think you divided by $\frac{4}{\pi}$, whereas, since you flipped it, you should have multiplied.

3. 3 things fundamentally wrong with that.

1) The minimum value of sin(x) on $\left[\frac{\pi}{4},\frac{\pi}{2}\right]$ is in the neighborhood of 0.7 This definitely suggests that the AVERAGE value is unlikely to be less than 0.7.

2) Square Units? Why would it be "square"? The sine function does not produce square units.

3) You showed NONE of your work so now no one can help you fix whatever it was that caused you to wander off.

Try this:
A) What is the total area under the curve?
B) If it were a rectangle with that width and that area, how tall would it be?

4. Originally Posted by VonNemo19
No, I'm sorry
At what point did I mess up. I'll explain how I did the problem

1/b-a, and got the anti derivative of sin x to be -cos x

pi/4(-cos x)

-pi/4(cos pi/2) -cos pi/3)
-pi/4(0-.7071)
+.5554 square units

At what point did I mess up. I'll explain how I did the problem

1/b-a, and got the anti derivative of sin x to be -cos x

pi/4(-cos x)

-pi/4(cos pi/2) -cos pi/3)
-pi/4(0-.7071)
+.5554 square units
$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\sin{x}dx\appr ox{.707}$

Now I've gotta divide by $b-a=\frac{\pi}{4}$. But dividing by $\frac{\pi}{4}$ is the same as multiplying by $\frac{4}{\pi}$.

$\approx\frac{4}{\pi}\cdot(.707)$

6. Originally Posted by VonNemo19
$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\sin{x}dx\appr ox{.707}$

Now I've gotta divide by $b-a=\frac{\pi}{4}$. But dividing by $\frac{\pi}{4}$ is the same as multiplying by $\frac{4}{\pi}$.

$\approx\frac{4}{\pi}\cdot(.707)$
Wouldn't it be -.707? I say this because [Cos(pi/2) - Cos (pi/4)] = -.707

Wouldn't it be -.707? I say this because [Cos(pi/2) - Cos (pi/4)] = -.707
But recall $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\sin x\,dx=\left.\left[{\color{red}-}\cos x\right]\right|_{\frac{\pi}{4}}^{\frac{\pi}{2}}$...

Thus, the average value would be $\frac{4}{\pi}\cdot\frac{\sqrt{2}}{2}=\frac{2\sqrt{ 2}}{\pi}$ (Its usually best to leave the answer as an exact value)

8. Originally Posted by Chris L T521
(Its usually best to leave the answer as an exact value)
I was gonna say something to that effect, but...

I forgot. Thanks chris.

9. Originally Posted by Chris L T521
But recall $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\sin x\,dx=\left.\left[{\color{red}-}\cos x\right]\right|_{\frac{\pi}{4}}^{\frac{\pi}{2}}$...

Thus, the average value would be $\frac{4}{\pi}\cdot\frac{\sqrt{2}}{2}=\frac{2\sqrt{ 2}}{\pi}$ (Its usually best to leave the answer as an exact value)
ahhh thank you both

10. is .9 the right answer?

11. You still don't seem to know why it is NOT "square units".

If you answer my two questions, you will KNOW and will not have to ask.

The correct answer is $\frac{2\sqrt{2}}{\pi}\approx 0.9003163158$

As I mentioned before, it is best that you leave the answer as an exact value, not an approximate value.

13. Originally Posted by Chris L T521
But recall $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\sin x\,dx=\left.\left[{\color{red}-}\cos x\right]\right|_{\frac{\pi}{4}}^{\frac{\pi}{2}}$...)
Maybe if I finish this thought....

$\left[-cos{x}\right]^{\frac{\pi}{2}}_{\frac{\pi}{4}}=\overbrace{-\cos{\frac{\pi}{2}}-[-(\cos{\frac{\pi}{4})}]=-0+\frac{\sqrt{2}}{2}}^{\text{We know these values because these are special angles}}=\frac{\sqrt{2}}{2}$

Multiply by $\frac{4}{\pi}$

14. Originally Posted by VonNemo19
Maybe if I finish this thought....

$\left[-cos{x}\right]^{\frac{\pi}{2}}_{\frac{\pi}{4}}=-\cos{\frac{\pi}{2}}-[-(\cos{\frac{\pi}{4})}]=-0+\frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{2}$

Multiply by $\frac{4}{\pi}$
Thank you but I know that you get 4/pi, but I don't understand how -cos (pi/4) = $=\frac{\sqrt{2}}{2}$ My calculator spits out .707 and i multiply it buy 4/pi and get .9

Thank you but I know that you get 4/pi, but I don't understand how -cos (pi/4) = $=\frac{\sqrt{2}}{2}$ My calculator spits out .707 and i multiply it buy 4/pi and get .9
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