MHB Integration with trig identities and absolute value

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In integration, we are allowed to use identities such as $$sinx = \sqrt{1-cos^2x}$$. Why does that work, and why doesn't make a difference in integration? Graphing $$\sqrt{1-cos^2x}$$ is only equal to sinx on certain intervals such as$$ (0, \pi) $$and $$(2\pi, 3\pi)$$. More correctly, shouldn't we use the absolute value of $$\sin\left({x}\right)$$?

$$sin^2x = 1 - cos^2x$$
$$|sinx| = \sqrt{1 = cos^2x}$$
or defined piecewisely = {$$\sin\left({x}\right)$$ in regions where it is above the x-axis, and -$$\sin\left({x}\right)$$ in regions where x is below the x-axis.

Is it possible to even truly isolate "$$\sin\left({x}\right)$$" from
$$sin^2x = 1 - cos^2x$$? It seems as the |$$\sin\left({x}\right)$$| is the closest we can to isolate it.

Sorry if I may seem confusing, but the concept of absolute value still confuses me.
 
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Rido12 said:
In integration, we are allowed to use identities such as $$sinx = \sqrt{1-cos^2x}$$. Why does that work, and why doesn't make a difference in integration? Graphing $$\sqrt{1-cos^2x}$$ is only equal to sinx on certain intervals such as$$ (0, \pi) $$and $$(2\pi, 3\pi)$$. More correctly, shouldn't we use the absolute value of $$\sin\left({x}\right)$$?

$$sin^2x = 1 - cos^2x$$
$$|sinx| = \sqrt{1 = cos^2x}$$
or defined piecewisely = {$$\sin\left({x}\right)$$ in regions where it is above the x-axis, and -$$\sin\left({x}\right)$$ in regions where x is below the x-axis.

Is it possible to even truly isolate "$$\sin\left({x}\right)$$" from
$$sin^2x = 1 - cos^2x$$? It seems as the |$$\sin\left({x}\right)$$| is the closest we can to isolate it.

Sorry if I may seem confusing, but the concept of absolute value still confuses me.

Hi Rido12! :)

As you surmise, we cannot just use $$\sin x = \sqrt{1-\cos^2x}$$.
Depending on the value of x, it can also be $$\sin x = -\sqrt{1-\cos^2x}$$

If we use the expression in an integral, we need to split the integral in separate integrals and use the appropriate version in each integral.

For instance:
$$\int_0^{2\pi} \sin x \, dx
= \int_0^{\pi} \sin x \, dx + \int_{\pi}^{2\pi} \sin x \, dx
= \int_0^{\pi} \sqrt{1-\cos^2x} \, dx + \int_{\pi}^{2\pi} -\sqrt{1-\cos^2x} \, dx$$
 
I like Serena said:
Hi Rido12! :)

As you surmise, we cannot just use $$\sin x = \sqrt{1-\cos^2x}$$.
Depending on the value of x, it can also be $$\sin x = -\sqrt{1-\cos^2x}$$

If we use the expression in an integral, we need to split the integral in separate integrals and use the appropriate version in each integral.

For instance:
$$\int_0^{2\pi} \sin x \, dx
= \int_0^{\pi} \sin x \, dx + \int_{\pi}^{2\pi} \sin x \, dx
= \int_0^{\pi} \sqrt{1-\cos^2x} \, dx + \int_{\pi}^{2\pi} -\sqrt{1-\cos^2x} \, dx$$

Thank you! My suspicions were correct. Btw, do you have any ideas as to why the absolute value is generally left out / ignored in integration, as in the following integral?

$$∫ x^5\sqrt{1-x^3} dx$$ , you can make the substitution $$ 1 - x^3 = z^2$$. The square root of $$z^2 $$ should then become |z|, but instead, it is just left as "z".
 
Last edited:
Rido12 said:
Thank you! My suspicions were correct. Btw, do you have any ideas as to why the absolute value is generally left out / ignored in integration, as in the following integral?

$$∫ x^5\sqrt{1-x^3} dx$$ , you can make the substitution $$ 1 - x^3 = z^2$$. The square root of $$z^2 $$ should then become |z|, but instead, it is just left as "z".

It's a choice.

We can pick $z$ either positive or negative, but not both.
The reason is that $1-x^3$ is a function that is monotone between 0 and infinity.
That means we have to pick $z$ such that $z^2$ is also monotone between 0 and infinity.
That is only the case if we either pick $z$ to be negative or we pick $z$ to be positive.

Calculating with a positive $z$ is usually easier, so that choice is often implicitly made.
 
What do you mean by "monotone"? You mean that it's like an even function? And that the interval (0, infinity) is the same as (-infinity, 0)?
 
Rido12 said:
Thank you! My suspicions were correct. Btw, do you have any ideas as to why the absolute value is generally left out / ignored in integration, as in the following integral?

$$∫ x^5\sqrt{1-x^3} dx$$ , you can make the substitution $$ 1 - x^3 = z^2$$. The square root of $$z^2 $$ should then become |z|, but instead, it is just left as "z".

While you probably could use that substitution, a better one (due to lack of ambiguity) is the substitution $\displaystyle \begin{align*} z = 1 - x^3 \implies \mathrm{d}z = -3x^2 \end{align*}$, giving

$\displaystyle \begin{align*} \int{ x^5 \, \sqrt{1 - x^3}\,\mathrm{d}x } &= - \frac{1}{3} \int{ x^3 \,\sqrt{1 - x^3} \, \left( -3x^2 \right) \, \mathrm{d}x } \\ &= -\frac{1}{3}\int{ \left( 1 - z \right) \, \sqrt{ z } \, \mathrm{d}z } \\ &= -\frac{1}{3} \int{ z^{\frac{1}{2}} - z^{\frac{3}{2}}\,\mathrm{d}z} \end{align*}$

and I'm sure you can go from here without any worry :)
 
Rido12 said:
What do you mean by "monotone"? You mean that it's like an even function? And that the interval (0, infinity) is the same as (-infinity, 0)?

Monotone means that the function passes the horizontal line test, which is pretty much required to switch to a different variable.

Both $1-x^3$ and $z^2$ have to pass this horizontal line test.
Furthermore, both need to be non-negative, since we need to be able to put them under a square root.

It means the $x$ is restricted to $(-\infty, 1]$ and that $z$ is restricted to either $(-\infty, 0]$ or $[0,\infty)$.To make it more explicit, if we evaluate the definite integral:
$$\int_{-2}^{1} x^5\sqrt{1-x^3} dx$$
We would get:
$$1-x^3=z^2 \quad \Rightarrow\quad x = (1-z^2)^{1/3} \quad \Rightarrow \quad dx = -\frac 2 3 z(1-z^2)^{-2/3} dz$$
If we pick $z \ge 0$ this becomes:
$$\int_{-2}^{1} x^5\sqrt{1-x^3} dx = \int_3^0 (1-z^2)^{5/3} \sqrt{z^2} \cdot -\frac 2 3 z(1-z^2)^{-2/3} dz = -\frac 2 3 \int_3^0 z(1-z^2) \sqrt{z^2}dz$$
Note that the direction of integration flips (from 3 down to 0) and also that a minus sign is introduced that effectively cancels that.

Since $z \ge 0$, we can now replace $\sqrt{z^2}$ by $z$ and we get:
$$-\frac 2 3 \int_3^0 z(1-z^2) \sqrt{z^2}dz = -\frac 2 3 \int_3^0 z(1-z^2) \cdot z\, dz$$
 
I understand all of it, except,
I like Serena said:
Note that the direction of integration flips (from 3 down to 0) and also that a minus sign is introduced that effectively cancels that.

Are you referring to if $$z \le 0$$, then the direction of integration flips as well as an extra minus sign that cancels with it?
 
Rido12 said:
I understand all of it, except,Are you referring to if $$z \le 0$$, then the direction of integration flips as well as an extra minus sign that cancels with it?

If we pick $z\le 0$ the direction of integration does not flip.
It will be from -3 up to 0.
 
  • #10
You said above that if $$z \ge 0$$, then
$$ -\frac 2 3 \int_3^0 z(1-z^2) \sqrt{z^2}dz = -\frac 2 3 \int_3^0 z(1-z^2) \cdot z\, dz$$

Then if $$z \le 0$$, wouldn't it be
$$ -\frac 2 3 \int_3^0 z(1-z^2) \sqrt{z^2}dz = -\frac 2 3 \int_3^0 z(1-z^2) \cdot -z\, dz $$ = $$\frac{2}{3} \int_3^0 z^2 (1-z^2)$$

Don't the two represent different areas? With the top being a positive quantity and the bottom being a negative?

By the way, I really appreciate the help! I love this forum, everyone here is incredibly helpful :)
 
  • #11
Rido12 said:
You said above that if $$z \ge 0$$, then
$$ -\frac 2 3 \int_3^0 z(1-z^2) \sqrt{z^2}dz = -\frac 2 3 \int_3^0 z(1-z^2) \cdot z\, dz$$

Then if $$z \le 0$$, wouldn't it be
$$ -\frac 2 3 \int_3^0 z(1-z^2) \sqrt{z^2}dz = -\frac 2 3 \int_3^0 z(1-z^2) \cdot -z\, dz $$ = $$\frac{2}{3} \int_3^0 z^2 (1-z^2)$$

Don't the two represent different areas? With the top being a positive quantity and the bottom being a negative?

With $z \le 0$, the lower boundary should be -3 instead of 3.
Combined with the fact that $\int_a^b f(x)dx = -\int_b^a f(x)dx$ they come out the same.

By the way, I really appreciate the help! I love this forum, everyone here is incredibly helpful :)

Thanks! ;)
 
  • #12
Rido12 said:
What do you mean by "monotone"? You mean that it's like an even function? And that the interval (0, infinity) is the same as (-infinity, 0)?

To clarify, from wikipedia, the rule for substitution is:

Let $I ⊆ ℝ$ be an interval and $\phi: [a,b] → I$ be a continuously differentiable function.
Suppose that $f : I → ℝ$ is a continuous function. Then:
$$
\int_{\phi(a)}^{\phi(b)} f(x)\,dx = \int_a^b f(\phi(t))\phi'(t)\, dt.
$$


In your case, you want to replace $1-x^3$ by $z^2$.
Or more specifically, replace $\sqrt{1-x^3}$ by either $z$ or by $-z$.
Now the function $\phi$ is supposed to map $z$ to $x$.
For that to happen, we need that the function $x \mapsto \sqrt{1-x^3}$ is invertible and the inverse function must be continuously differentiable.
A typical way to achieve that is when the function is strictly monotone and differentiable.

It means that you would pick for instance:
$$\phi: z \mapsto (1-z^2)^{1/3}$$
and
$$f: x \mapsto x^5\sqrt{1-x^3}$$
after which you can apply the substitution rule as mentioned on wiki.
 
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