Use the washer method to determine the volume of the solid formed when the region bounded by y=1/2x^2, y=0, and x=2 is rotated about the line y=3.
The curves intersect at (2,2)
So, using washers of thickness dx
v = ∫[0,2] π(R^2-r^2) dx
where R = 3-y = 3 - 1/2 x^2 and r= 3-2 = 1
v = ∫[0,2] π((3 - 1/2 x^2)^2-1^2) dx = 48π/5
to check your answer, use shells of thickness dy
v = ∫[0,2] 2πrh dy
where r=3-y and h = x = √(2y)
v = ∫[0,2] 2π(3-y)√(2y) dy = 48π/5
To use the washer method to determine the volume of the solid, follow these steps:
1. Visualize the region: Sketch the region bounded by the equations y = 1/2x^2, y = 0, and x = 2 on a graph. This will help you understand the shape of the solid formed when the region is rotated.
2. Determine the outer radius: The outer radius of each washer is the distance from the rotation axis (y = 3) to the outer edge of the region. In this case, the outer radius is simply 3 minus the value of y = 1/2x^2. Therefore, the outer radius is given by: R = 3 - (1/2x^2).
3. Determine the inner radius: The inner radius of each washer is the distance from the rotation axis (y = 3) to the inner edge of the region. In this case, the inner radius is simply 3 minus the value of y = 0. Therefore, the inner radius is given by: r = 3 - 0.
4. Integrate the volume: The volume of each washer is given by the formula V = π(R^2 - r^2), where R is the outer radius, and r is the inner radius. Integrate this formula with respect to x over the interval [0, 2] to find the total volume:
V = ∫[0,2] π((3 - (1/2x^2))^2 - (3 - 0)^2) dx
5. Evaluate the integral: Solve the integral to find the volume of the solid.
V = ∫[0,2] π(9 - 3(1/2x^2) + 1/4x^4 - 9) dx
Simplifying this expression will yield the final volume.
Note: Calculating the integral requires knowledge of calculus techniques, such as integration rules and methods. If you are not familiar with calculus, consider consulting a mathematical software or using an online calculator that can efficiently solve definite integrals.
To find the volume of the solid formed when the region bounded by y = 1/2x^2, y = 0, and x = 2 is rotated about the line y = 3 using the washer method, follow these steps:
Step 1: Sketch the region
Draw a graph of the given functions and lines to visualize the region bounded by y = 1/2x^2, y = 0, and x = 2. The region will have a closed top formed by the parabola and a closed bottom formed by the x-axis:
┌───────╲
│ ╲
│ ╲
│ ╲
│ ╲
│ π[R(x)^2-r(x)^2]dx
│ ╲
│ ╲
│ ╲
│ ╲
│y=3 ╲
└───────╲───────────
y=0 x=2
Step 2: Determine the outer radius (R(x))
The outer radius (R(x)) is the perpendicular distance from the line of rotation (y = 3) to the curve. In this case, the line y = 3 is above the parabola. So, the outer radius can be found by subtracting the equation of the parabola from the equation of the line:
R(x) = 3 - (1/2)x^2
Step 3: Determine the inner radius (r(x))
The inner radius (r(x)) is the perpendicular distance from the line of rotation (y = 3) to the x-axis. Since the region is bounded by the x-axis, the inner radius is simply the constant distance between the line of rotation and the x-axis:
r(x) = 3 - 0 = 3
Step 4: Set up the integral
The washer method involves integrating the difference in the areas of the outer and inner disks formed when the region is rotated. The volume can be calculated using the following integral:
V = ∫[a, b] π[R(x)^2 - r(x)^2] dx
In this case, a = 0 (the leftmost boundary of the region) and b = 2 (the rightmost boundary of the region). Substituting the values for R(x) and r(x):
V = ∫[0, 2] π[(3 - (1/2)x^2)^2 - 3^2] dx
Simplifying the expression inside the integral:
V = ∫[0, 2] π[(9 - 3x^2 + (1/4)x^4) - 9] dx
= ∫[0, 2] π(1/4)x^4 - (3/2)x^2 dx
Step 5: Evaluate the integral
Integrate the expression with respect to x:
V = π(1/4)∫[0, 2]x^4 dx - π(3/2)∫[0, 2]x^2 dx
Integrating each term separately:
V = π(1/4)(1/5)x^5 ∣ [0, 2] - π(3/2)(1/3)x^3 ∣ [0, 2]
Substituting the limits:
V = π(1/4)(1/5)(2^5) - π(3/2)(1/3)(2^3)
= π(1/4)(1/5)(2^5) - π(3/2)(1/3)(2^3)
Calculating the values:
V = π(32/20) - π(24/6)
= π(8/5) - π(8)
= (8/5 - 8)π
= (8 - 40/5)π
= (40/5 - 40/5)π
= 0π
= 0
Step 6: Finalize the answer
The volume of the solid formed when the region bounded by y = 1/2x^2, y = 0, and x = 2 is rotated about the line y = 3 using the washer method is 0 cubic units.