The rectangle ABCD is rotated about the side AB. Find the volume of the solid defined by this rotation.
AB = DC = 4cm
AD = CB = 2cm
What you get when you perform the indicated rotation of the rectangle about side AB is a cylinder with radius = 2 cm and length h = 4 cm. The line AB is the axis of the cylinder.
The volume is
V = pi*r^2*h = 16*pi cm^3
Do the calculation.
I've got it!
Thank you very much for your help!
The real problem is that I do not comprehend the question.
The part I'm really stumbling over is the "rotated about the side AB". I have no idea what that means..
I have another question that is quite similar...so if anyone can explain, it would be greatly appreciated.
To find the volume of the solid defined by rotating the rectangle ABCD about side AB, we can use the method of cylindrical shells.
Cylindrical shells are formed by taking thin cylindrical slices of the solid and finding their volume. Here's how we can approach this problem:
1. Visualize the solid: The rotation of the rectangle ABCD about side AB creates a cylinder. The height of the cylinder will be equal to the length of AB, which is 4 cm. The circumference of the cylinder will be equal to the perimeter of the rectangle, which is 2(AB + AD) = 2(4 + 2) = 12 cm.
2. Determine the radius of the cylinder: The radius will be equal to the distance from the axis of rotation (side AB) to any point on the rectangle. In this case, the distance from the axis of rotation to side AD (or side CB) is 2 cm.
3. Set up the integral: We need to find the volume by summing up the volumes of all the cylindrical shells. The volume of each shell can be calculated as the product of its height (cylinder height) and the circumference of the circle formed by the shell. We can integrate the volume over the interval [0, 4] (since the height of the cylinder is 4 cm).
4. Calculate the volume: Using the formula for the volume of a cylindrical shell, we have:
V = ∫[0,4] 2πrh dr
where r is the radius (2 cm) and h is the height (4 cm).
V = ∫[0,4] 2π(2)(4) dr
V = 16π ∫[0,4] dr
V = 16π [r] from 0 to 4
V = 16π (4 - 0)
V = 64π cm^3
Therefore, the volume of the solid defined by rotating the rectangle ABCD about side AB is 64π cm^3.
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