the volume generated by rotating the region about y= 2 is
To find the volume generated by rotating the region bounded by the curves x = = 7y²,
y ≥ 0, x = 7 about the axis y = 2, we can use the method of cylindrical shells. The formula for the volume using cylindrical shells is given by:
V = 2π fxf(x) dx
where f(x) is the distance from the axis of rotation to the outer curve at x-
coordinate x, and [a, b] is the interval along the x-axis.
In this case, the outer curve is x = 7 and the inner curve is a
=
rotation is y
shell is x.
7y². The axis of
=
2. So, the radius of the cylindrical shell is 2-y, and the height of the
First, let's find the intersection points of the two curves by setting them equal to each
other:
7y² = 7
Solving for y:
y² = 1
y = 1
So, the bounds of integration are y = 0 to y = 1.
So, the bounds of integration are y = 0 to y = 1.
Now, we express x in terms of y:
x = 7y²
Now, we can set up the integral:
V = 2π S¹ (7y²) (7 — y) dy
Simplify the integrand:
V = 2π S¹ (49y² – 7y³) dy
Integrate term by term:
V = 2π [±ºy³ – ²y¹] |
Evaluate at the upper and lower limits:
V = 2T (4⁹)
Now, calculate the final result:
V = 2T (147-21)
12
V = π (49 - 21)
V = π-28
V = 5
So, the volume generated by rotating the region about y= 2 is
π cubic units.