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How do you find the volume of a rotated region bounded by \[y=\sqrt{x},y=3\], the \[y\] - axis about the \[y\] - axis?

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Last updated date: 24th Feb 2024
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Hint: From the question given, we have been asked to find the volume of a rotated region bounded by \[y=\sqrt{x},y=3\], the \[y\] - axis about the \[y\] - axis. To solve the given question, we have to draw the pictorial representation that is the graph for the given question. By using that we can solve the question given.

Complete step by step answer:
Pictorial representation for the given question is shown below:
seo images


By using the above graph, we have to find the volume bounded by the region.
We know that, circular cross sections of the bounded region have an area \[\pi {{x}^{2}}\]
Or, since \[x={{y}^{2}}\]
\[\Rightarrow A\left( y \right)=\pi {{y}^{4}}\]
For a thin enough slice, \[\Delta y\], the volume of the slice approaches \[\Rightarrow S\left( y \right)=\Delta y.A\left( y \right)\]
And the volume of the bounded region will be \[\Rightarrow V\left( y \right)=\int\limits_{0}^{3}{\pi {{y}^{4}}dy}\]
Therefore, we have to integrate the above equation to get the volume of the bounded region.
By integrating and simplifying the above equation, we get
\[\Rightarrow V\left( y \right)=\pi \int\limits_{0}^{3}{{{y}^{4}}dy}\]
\[\Rightarrow V\left( y \right)=\pi \dfrac{{{y}^{5}}}{5}|_{0}^{3}\]
\[\Rightarrow V\left( y \right)=48.6\pi \]
Therefore, we got the volume of the bounded region.

Note: We should be very careful while drawing the graph that is the pictorial representation of the given question. Also, we should be well aware of the integration concept. Also, we should be very careful while doing the calculation of the integration part of the above problem. Also, we should be well known about the limits that have to be taken to the integration. We should be very careful while applying the limits for the integration. Also, we should use the graph to find the limits for the integration. Similarly we can use integration and differentiation to find the volume or area of any curves which is not specified before.