# Surface area of rotation about the y-axis

#### Petrus

##### Well-known member
Calculate the area of ​​the surface of rotation which occurs when the curve
rotate in y-axe.
I start with $$\displaystyle x=\sqrt{28y}$$ then $$\displaystyle f'(x)=\frac{14}{\sqrt{28y}}$$
so we got
$$\displaystyle 2\pi\int_0^{21}\sqrt{28y}\sqrt{1+(\frac{14}{\sqrt{28y}})}^2$$
then I rewrite as $$\displaystyle \int_0^2\sqrt{28y}\sqrt{1+\frac{196}{28y}}$$
$$\displaystyle \sqrt{28y}\sqrt{1+\frac{196}{28y}}<=>\sqrt{28y+196}$$
So I got $$\displaystyle 2\pi\int_0^{21}\sqrt{28y+196}$$ and if Integrate it I get
$$\displaystyle \frac{(28y+196)^{1.5}}{1.5*28}$$ Is this correct?

#### MarkFL

Staff member
You have the right idea, but you want to write things in terms of $x$.

$$\displaystyle S=2\pi\int_a^b x\sqrt{1+\left[f'(x) \right]^2}\,dx$$

You want the lower limit of integration to be the $x$-value that corresponds with $y=0$, and the upper limit to correspond with $y=21$.

If $$\displaystyle f(x)=\frac{x^2}{28}$$, then what is $f'(x)$?

edit: I see now that your approach is valid, and easier to integrate.

Your anti-derivative is correct (except don't forget the $2\pi$), so apply the FTOC to find the value of the definite integral.

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#### Petrus

##### Well-known member
You have the right idea, but you want to write things in terms of $x$.

$$\displaystyle S=2\pi\int_a^b x\sqrt{1+\left[f'(x) \right]^2}\,dx$$

You want the lower limit of integration to be the $x$-value that corresponds with $y=0$, and the upper limit to correspond with $y=21$.

If $$\displaystyle f(x)=\frac{x^2}{28}$$, then what is $f'(x)$?

edit: I see now that your approach is valid, and easier to integrate.
$$\displaystyle \frac{x^3}{84}$$

#### MarkFL

Staff member
$$\displaystyle \frac{x^3}{84}$$
You integrated, rather than differentiated, but disregard what I posted about that, your method is correct and easier to use.

#### Petrus

##### Well-known member
You integrated, rather than differentiated, but disregard what I posted about that, your method is correct and easier to use.
So the answer is $$\displaystyle 2\pi*\frac{(28*21+196)^{1.5}}{1.5*28}$$ I am correct?

#### MarkFL

Staff member
No, you need to evaluate:

$$\displaystyle 2\pi\left[\frac{(28y+196)^{\frac{3}{2}}}{42} \right]_0^{21}=\frac{\pi}{21}\left((28\cdot21+196)^{\frac{3}{2}}-(28\cdot0+196)^{\frac{3}{2}} \right)=?$$

#### Petrus

##### Well-known member
No, you need to evaluate:

$$\displaystyle 2\pi\left[\frac{(28y+196)^{\frac{3}{2}}}{42} \right]_0^{21}=\frac{\pi}{21}\left((28\cdot21+196)^{\frac{3}{2}}-(28\cdot0+196)^{\frac{3}{2}} \right)=?$$
$$\displaystyle \frac{\pi}{21}((28*21+196)^{\frac{3}{2}}-196^{\frac{3}{2}})$$ I don't wanna use any calculator so that would be my answer.

#### MarkFL

Staff member
Without using a calculator, you could find:

$$\displaystyle 28\cdot21+196=4(3\cdot7^2+49)=16\cdot49=28^2$$

and so:

$$\displaystyle (28^2)^{\frac{3}{2}}=28^3$$

and then:

$$\displaystyle 28\cdot0+196=14^2$$

and then you have:

$$\displaystyle S=\frac{\pi}{21}(28^3-14^3)=\frac{14^3\pi}{21}(2^3-1)=\frac{14^3\pi}{3}=\frac{2744\pi}{3}$$

all easily done without a calculator.

#### Petrus

##### Well-known member
Without using a calculator, you could find:

$$\displaystyle 28\cdot21+196=4(3\cdot7^2+49)=16\cdot49=28^2$$

and so:

$$\displaystyle (28^2)^{\frac{3}{2}}=28^3$$

and then:

$$\displaystyle 28\cdot0+196=14^2$$

and then you have:

$$\displaystyle S=\frac{\pi}{21}(28^3-14^3)=\frac{14^3\pi}{21}(2^3-1)=\frac{14^3\pi}{3}=\frac{2744\pi}{3}$$

all easily done without a calculator.
hmm I get wrong with FTOC
If we subsitute $$\displaystyle u=28y+196$$ then we get$$\displaystyle \frac{\pi}{4}\int_{196}^{784}\sqrt{28y+196}$$

edit: I am confused with this change of 2pi. Some does subsitute and then change that 2pi and other does not. I have been serching over internet and don't understand that.

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#### MarkFL

Staff member
hmm I get wrong with FTOC
If we subsitute $$\displaystyle u=28y+196$$ then we get$$\displaystyle \frac{\pi}{4}\int_{196}^{784}\sqrt{28y+196}$$

edit: I am confused with this change of 2pi. Some does subsitute and then change that 2pi and other does not. I have been serching over internet and don't understand that.
Using the substitution you cite, $$\displaystyle u=28y+196$$, we get:

$$\displaystyle S=\frac{2\pi}{28}\int_{28\cdot0+196}^{28\cdot21+196}u^{\frac{1}{2}}\,du$$

$$\displaystyle S=\frac{\pi}{14}\int_{196}^{784}u^{\frac{1}{2}}\,du$$

$$\displaystyle S=\frac{\pi}{21}\left[u^{\frac{3}{2}} \right]_{196}^{784}=\frac{\pi}{21}(28^3-14^3)=\frac{2744\pi}{3}$$

#### Petrus

##### Well-known member
Using the substitution you cite, $$\displaystyle u=28y+196$$, we get:

$$\displaystyle S=\frac{2\pi}{28}\int_{28\cdot0+196}^{28\cdot21+196}u^{\frac{1}{2}}\,du$$

$$\displaystyle S=\frac{\pi}{14}\int_{196}^{784}u^{\frac{1}{2}}\,du$$

$$\displaystyle S=\frac{\pi}{21}\left[u^{\frac{3}{2}} \right]_{196}^{784}=\frac{\pi}{21}(28^3-14^3)=\frac{2744\pi}{3}$$
Thanks again Mark Now I understand! Thanks for the $$\displaystyle \LaTeX$$ code as well