High School Definite integrals with +ve and -ve values

[HuskyLab]

I understand that if you have a function in which you want to determine the full (i.e. account for positive and negative values) integral you need to break down your limits into separate intervals accordingly.

Is there any way in which you can avoid this or is it mathematically impossible? If so, can someone explain to me why you can't?

I thought about trying to incorporate a rectified sine wave (abs(sinx) as an alternative but is such a function applicable/usable (is it even considered continuous)? How do you represent and integrate such a function algebraically?

Thanks.

 

[andrewkirk]

It depends what you mean by the full integral.

Perhaps what you are searching for is the RMS (root mean square) value, which is
$$\sqrt{\int_a^b f(x)^2\,dx}$$
This is used for instance as a measure of average voltage and current in AC circuits, since the simple integral will be close to zero because the positives and negatives cancel each other out.

 

[HuskyLab]

It depends what you mean by the full integral.

Perhaps what you are searching for is the RMS (root mean square) value, which is
$$\sqrt{\int_a^b f(x)^2\,dx}$$
This is used for instance as a measure of average voltage and current in AC circuits, since the simple integral will be close to zero because the positives and negatives cancel each other out.

So squaring the function will rectify it and then the subsequent values calculated in the integral of interval a->b then rooted will be the same as:
$$\int_a^b |sin(x)|\,dx$$

Is that correct?

So I just calculated it myself and they are not equivalent.

$$\int_a^b |sin(x)|\,dx \neq \sqrt{\int_a^b f(x)^2\,dx}$$

 

[Mark44]

So squaring the function will rectify it and then the subsequent values calculated in the integral of interval a->b then rooted will be the same as:
$$\int_a^b |sin(x)|\,dx$$

Is that correct?

So I just calculated it myself and they are not equivalent.

$$\int_a^b |sin(x)|\,dx \neq \sqrt{\int_a^b f(x)^2\,dx}$$

But these are equal:
$$\int_a^b |sin(x)|\,dx = \int_a^b \sqrt{\sin^2(x)}\,dx$$

for the reason that $|x| = \sqrt{x^2}$, by definition.

 

[HuskyLab]

But these are equal:
$$\int_a^b |sin(x)|\,dx = \int_a^b \sqrt{\sin^2(x)}\,dx$$

for the reason that $|x| = \sqrt{x^2}$, by definition.

Yeah, that makes sense. Do you know of any way in which you can calculate the total area under a sine function without breaking down the limits or does the question itself not make sense? Integrating something akin to a rectified sine wave? $$|sin(x)|$$

 

[Mark44]

Yeah, that makes sense. Do you know of any way in which you can calculate the total area under a sine function without breaking down the limits or does the question itself not make sense? Integrating something akin to a rectified sine wave? $$|sin(x)|$$

It depends on whether the question asks for the integral, say, $\int_a^b \sin(x)dx$ or the area between the graph of y = sin(x) and the x-axis between x = a and x = b. If the question asks for area, you need to consider the intervals where the graph lies above the x-axis and those intervals where the graph is below. Changing to absolute value doesn't seem like a reasonable shortcut to me, as you still need to replace |sin(x)| by sin(x) on the intervals where sin(x) >= 0, and to -sin(x) on the intervals where sin(x) < 0.