The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
y = −x^2 + 14x − 45, y = 0; about the x-axis
To find the volume of the resulting solid when the given region is rotated about the x-axis, we can use the method of cylindrical shells.
Step 1: Determine the bounds of integration
To find the bounds of integration, we need to find the x-values at which the curves intersect.
y = −x^2 + 14x − 45 = 0
Using factoring or the quadratic formula, we can find that the roots of this equation are x = 5 and x = 9.
So, the bounds of integration will be from x = 5 to x = 9.
Step 2: Determine the height of each cylindrical shell
The height of each cylindrical shell is given by the y-value of the curve −x^2 + 14x − 45.
So, the height (h) of each cylindrical shell will be h = −x^2 + 14x − 45.
Step 3: Determine the radius of each cylindrical shell
The radius (r) of each cylindrical shell is given by the x-value at which the curve intersects the x-axis (y = 0).
So, the radius (r) of each cylindrical shell will be r = x.
Step 4: Determine the volume of each cylindrical shell
The volume (dV) of each cylindrical shell is given by the formula:
dV = 2πrh dx
Replacing the values of height (h) and radius (r), we get:
dV = 2π(x)(−x^2 + 14x − 45) dx
Step 5: Integrate to find the total volume
To find the total volume, we need to integrate the above expression from the lower bound (x = 5) to the upper bound (x = 9):
V = ∫[5 to 9] 2π(x)(−x^2 + 14x − 45) dx
Evaluating this integral will give us the volume (V) of the resulting solid.
To find the volume of the solid formed by rotating the region bounded by the curves y = −x^2 + 14x − 45 and y = 0 about the x-axis, we can use the method of cylindrical shells.
The idea behind this method is to consider an infinitesimally thin strip along the x-axis and rotate it around the axis of rotation (in this case, the x-axis) to form a cylindrical shell. The volume of each cylindrical shell can be calculated and then integrated over the range of x-values that define the region.
Here are the steps to find the volume using the cylindrical shell method:
1. Determine the limits of integration. In this case, we need to find the x-values where the two curves intersect.
Set the equation y = −x^2 + 14x − 45 to 0 and solve for x:
−x^2 + 14x − 45 = 0
Factor the quadratic equation:
(x − 5)(x − 9) = 0
This gives us two x-values: x = 5 and x = 9.
Therefore, the limits of integration will be from x = 5 to x = 9.
2. Express the volume of a cylindrical shell. The volume of a cylindrical shell is given by the formula:
V = 2πrhΔx
In this formula, r represents the radius of the shell, h represents the height of the shell (corresponding to the difference in y-values between the curves), and Δx represents the thickness of the shell along the x-axis.
3. Express the radius and height in terms of x. In this case, since we are rotating about the x-axis, the radius of each shell will be the distance from the x-axis to the curve y = −x^2 + 14x − 45, which is the positive value of y.
r = |y| = −y = x^2 − 14x + 45
The height of each shell will be the difference between the curves:
h = y_top − y_bottom = 0 − (−x^2 + 14x − 45) = x^2 − 14x + 45
4. Calculate the volume of each cylindrical shell. Substitute the expressions for r, h, and Δx into the volume formula:
V = ∫[a, b] 2π(x^2 − 14x + 45)(x^2 − 14x + 45) dx
= 2π∫[a, b] (x^2 − 14x + 45)^2 dx
where [a, b] represents the interval of integration, in this case, [5, 9].
5. Integrate the volume formula. Evaluate the integral using appropriate integration techniques, such as expansion, substitution, or trigonometric substitution.
V = 2π ∫[5, 9] (x^2 − 14x + 45)^2 dx
This step requires advanced calculus techniques, and the evaluation depends on your familiarity with integration methods.
the curve intersects y=0 at x=5,9
v = ∫[5,9] π (−x^2 + 14x − 45)^2 dx
= π (x^5/5 - 7x^4 + 286/3 x^3 - 630x^2 + 2025x [5,9]
= 512/15 π