What is the volume of the solid formed by rotating the line y = 381 - 2499 around the y - axis?

The equation you provided, "y = 381 - 2499", simplifies to "y = -2118". This is a horizontal line. When we rotate a horizontal line around the y - axis, the solid formed is a cylinder.

Understanding the Rotation of a Line around the Y - Axis

Let's first understand the concept of rotating a curve around an axis to form a solid. When we rotate a curve (y = f(x)) around the y - axis from (x=a) to (x = b), we can use the method of cylindrical shells to find the volume of the resulting solid. The formula for the volume (V) using the cylindrical shells method is (V=2\pi\int_{a}^{b}x|f(x)|dx).

In our case, since (y=-2118) is a horizontal line, we can think of it in the context of the (x - y) plane. If we assume that we are considering the region in the (x - y) plane where (x) ranges from (x = 0) to some non - negative value (x = R). When we rotate the line (y=-2118) around the y - axis, the resulting solid is a cylinder.

The height (h) of the cylinder is the absolute value of the (y) - value of the line, (h = |- 2118|=2118), and the radius (r) of the base of the cylinder is determined by the range of (x). If we assume that the line extends from (x = 0) to (x = R), the volume (V) of a cylinder is given by the formula (V=\pi r^{2}h).

Calculating the Volume

Since the line (y = -2118) and we rotate it around the y - axis, if we assume that the line is considered from (x = 0) to (x = 1) (for the sake of simplicity, we can choose any non - negative range of (x), and the general principle remains the same), the radius (r = 1) and the height (h=2118).

Using the formula (V=\pi r^{2}h), we substitute (r = 1) and (h = 2118) into the formula. So, (V=\pi\times(1)^{2}\times2118 = 2118\pi\approx 2118\times3.14159=6653.89) cubic units.

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References

• Stewart, James. "Calculus: Early Transcendentals". Cengage Learning, 2015.
• Thomas, George B., et al. "Thomas' Calculus". Pearson, 2018.