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Help on a calculus problem involving the volume of a solid with known cross-sections.?
The problem is
"The solid with a semicircular base of raiud 5 whose cross sections perpendicular to the base and parallel to the diameter are squares".
I got an answer but it is exactly 1/2 of the correct answer.
With the description, the equation is y = sqrtOf(25 - x^2).
And since the cross sections are squares, it's y^2 = 25-x^2.
Taking the integral of this function I got 25x - (x^3/3). With function being even, I can evaluate the integral from 0 to 5 and multiply the result by 2.
Doing so I got 500/3 but the correct answer is 1000/3.
Is there a step I'm missing? because I can't seem to figure out why my answer is only half of the correct.
3 Answers
- ?Lv 74 years ago
First we find an equation for semi-circle in xy-plane:
x² + y² = 25, where −5 ≤ x ≤ 5, 0 ≤ y ≤ 5
Now what you did was take cross-sections perpendicular to x-axis. In other words, perpendicular to diameter, instead of parallel to diameter.
Since diameter runs parallel to x-axis, you need to take cross-sections perpendicular to y-axis. These cross-sections have side length = 2x, and
area = (2x)(2x) = 4x² = 4(25−y²)
This integrand is 4 times larger than the one you used.
But we only integrate from y = 0 and y = 5, whereas you integrated from x = 0 to x = 5 and multiplied by 2. So final result should be twice what you got.
∫ [0 to 5] 4(25−y²) dy = 1000/3
- ?Lv 74 years ago
The cross-sections are squares, but your squares are not parallel to the diameter as required.
