# Thread: volume of solid hemisphere

1. ## volume of solid hemisphere

Estimate the volume of a solid hemisphere of radius 3, imagining the axis of symmetry to be the x-axis. Partition the interval [0,3] into six subintervals of equal length and approximate the solid with cylinders based on the circular cross sections of the hemisphere perpendicular to the x-axis at the subintervals' left endpoints.

How would I start this problem?
Any help is appreciated!

2. Originally Posted by live_laugh_luv27
Estimate the volume of a solid hemisphere of radius 3, imagining the axis of symmetry to be the x-axis. Partition the interval [0,3] into six subintervals of equal length and approximate the solid with cylinders based on the circular cross sections of the hemisphere perpendicular to the x-axis at the subintervals' left endpoints.

How would I start this problem?
Any help is appreciated!

So we're using Left Hand Sums.

First find the height of each cylinder.

$\Delta x = \frac{b-a}{n} = \frac{3 - 0}{6} = 0.5
$

It's six subintervals, so we'll have six cylinders.

Note that the height function is the upper half of the circle of radius 3:

$y=\sqrt{9-x^2}$

The radii for the six cylinders are the y-values of the left end points:

[0, 1/2]
[1/2, 2/2]
[2/2, 3/2]
[3/2, 4/2]
[4/2, 5/2]
[5/2, 6/2]

The Volume formula is

$V_i = \pi r^2 h = \pi y^2 \Delta x$

for example:

$V_1 = \pi (f(o))^2 0.5 = \pi 9 * 0.5 = \frac{9 \pi}{2}$
$V_2 = \pi (f(1/2))^2 0.5 = \pi \frac{35}{4} * 0.5 = \frac{35 \pi}{8}$
....

Compute the remaining cylinders and add them up to get the approximation.

Good luck!!

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# to estimate the volume of a solid hemisphere

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