contestada

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = ln(3x), y = 1, y = 5, x = 0; about the y-axis

Respuesta :

LammettHash

Using the disk method, the volume is obtained with the integral

[tex]\displaystyle\pi\int_{y=1}^{y=5}\left(\frac{e^y}3\right)^2\,\mathrm dy=\frac\pi9\int_1^5e^{2y}\,\mathrm dy[/tex]

The radius of each disk is given by the horizontal distance from the axis of revolution, [tex]x=0[/tex], to the logarithmic curve which we can write as a [tex]x(y)[/tex]:

[tex]y=\ln(3x)\implies e^y=e^{\ln3x}=3x\implies x=\dfrac{e^y}3[/tex]

Then the volume of one such disk is [tex]\pi x(y)^2[/tex].

The volume of the entire solid would be

[tex]\displaystyle\frac\pi9\int_1^5e^{2y}\,\mathrm dy=\frac\pi{18}\int_1^52e^{2y}\,\mathrm dy=\frac\pi{18}\int_1^5e^{2y}\,\mathrm d(2y)[/tex]

[tex]=\dfrac\pi{18}\left(e^{10}-e^2\right)[/tex]

Ver imagen LammettHash
ajeigbeibraheem

The volume V of the solid obtained by rotating the region bounded by given curves is [tex]\mathbf{\dfrac{\pi}{18} \ e^2 \Big[e^8 -1 \Big]}[/tex]

From the information given that, the region of the curves is bounded by y = 1 and y = 5.

Thus, the volume V of the solid can be determined by using the integral formula:

[tex]\mathbf{V = \pi \int x^2 dy}[/tex]

[tex]\mathbf{V = \int ^{y = 5}_{y=1} \pi x^2 dy---(1)}[/tex]

where;

  • y = In(3x)

∴

By finding (x), we will make (x) the subject of the formula:

i.e.

[tex]\mathbf{3x = e^y}[/tex]

[tex]\mathbf{x = \dfrac{e^y}{3}}[/tex]

Now, replacing the value of (x) back into equation (1), we have:

[tex]\mathbf{V = \int ^{5}_{1} \pi (\dfrac{e^y}{3})^2 dy }[/tex]

[tex]\mathbf{V =\dfrac{\pi}{9} \int ^{5}_{1} e^{2y} dy }[/tex]

[tex]\mathbf{V =\dfrac{\pi}{9} \Big [\dfrac{e^{2y}}{2} \Big]^5_1}[/tex]

[tex]\mathbf{V =\dfrac{\pi}{18} \Big [e^{2(5)} - e^{2(1)} \Big]}[/tex]

[tex]\mathbf{V =\dfrac{\pi}{18} \ e^2\Big [e^{8} - 1 \Big]}[/tex]

Therefore, we can conclude that the volume V of the solid obtained by rotating the region bounded by given curves is [tex]\mathbf{\dfrac{\pi}{18} \ e^2 \Big[e^8 -1 \Big]}[/tex]

Learn more about region bounded by curves here:

https://brainly.com/question/4965843?referrer=searchResults

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