Question

Suppose That \( R \) Is The Finite Region Bounded By \( Y=X, Y=X+1, X=0 \), And \( X=3 \). find the exact value of the volume of the object we obtain when rotating about the -axis.

Answer 1

Explanation:

Find Object Volume Rotation

Suppose That \( R \) Is The Finite Region Bounded By \( Y=X, Y=X+1, X=0 \), And \( X=3 \). find the exact value of the volume of the object we obtain when rotating about the -axis.

The region R is a triangular region defined by the lines y = x, y = x + 1, x = 0, and x = 3. To find the volume of the object obtained by rotating this region about the x-axis, we can use the method of cylindrical shells.

The height of the cylindrical shell is given by the difference between y = x and y = x + 1, or 1 unit. The radius of the cylindrical shell is given by x. Therefore, the volume of the object is given by:

\begin{align*}

V &= \int_{x=0}^{x=3} \pi x^2 \cdot 1, dx \

&= \pi \int_{x=0}^{x=3} x^2, dx \

&= \pi \left[ \frac{x^3}{3} \right]_{x=0}^{x=3} \

&= \pi \left( \frac{27}{3} - \frac{0}{3} \right) \

&= 9\pi

\end{align*}

So the volume of the object is equal to 9π cubic units.