$Basic calc question$

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$Basic calc question$

Determine the volume generated by rotating about the X-axis the area between the function: 𝒚 = 𝟒𝒆𝒙/𝟓 between 𝒙 = 𝟐𝒎 and 𝒙 = 𝟓𝒎, sketch the resulting solid including x & y scales.

Calculus Precalculus

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The function that we have is<br />$$ y(x) = 4 e^{x/5} $$<br /><br />The volume defined by rotating the function with respect to the x-axis is given by<br /> $$ V = \int_{2}^5 \pi y(x)^2 dx = 16 \pi \int_{2}^5 e^{\frac{2}{5} x}dx = 16 \pi \left[ \frac{5}{2} e^{\frac{2 x}{5}} \right]_{2}^5 =40 \left(e^2-e^{4/5}\right) \pi \approx \blue{648.866 \left[ m^3 \right]}$$

In the file attached, find the sketch of the volume. The red curve corresponds to $y(x)$, and the purple circle represents the rotation about the x-axis.

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$Basic calc question$

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