A solid is 12 inches high. The cross-section of the solid at height x above its base has area 3x square inches. Find the volume of the solid.
How do you solve this problem? I don't have any boundaries, so I cannot find the are between two curves. So how do you find the limits of integration? How do you find the base? Do you utilize special triangle properties to find the base?
How do you solve this problem? I don't have any boundaries, so I cannot find the are between two curves. So how do you find the limits of integration? How do you find the base? Do you utilize special triangle properties to find the base?
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"I don't have any boundaries, so I cannot find the are between two curves."
To solve, you must recognize that there are two inaccuracies in that statement.
1) You DO have boundaries -- the solid is 12" high.
2) You CAN find the area -- in fact, it is given to you.
The area "at height x above its base has area 3x square inches."
Therefore the volume of any slice is ΔV = 3x*Δx
We sum them up through integration:
V = ∫[0,12] 3x dx = 3x²/2 |[0,12] = 216 units³
Note that we don't know the shape of the solid, only its height.
It might, though, be an upside-down square pyramid with base 3√6 and height 12.
Or a right circular cone with radius 3√(6/π) and height 12. Or ...
To solve, you must recognize that there are two inaccuracies in that statement.
1) You DO have boundaries -- the solid is 12" high.
2) You CAN find the area -- in fact, it is given to you.
The area "at height x above its base has area 3x square inches."
Therefore the volume of any slice is ΔV = 3x*Δx
We sum them up through integration:
V = ∫[0,12] 3x dx = 3x²/2 |[0,12] = 216 units³
Note that we don't know the shape of the solid, only its height.
It might, though, be an upside-down square pyramid with base 3√6 and height 12.
Or a right circular cone with radius 3√(6/π) and height 12. Or ...