Find the volume of the described solid $ S $.

The base of $ S $ is the triangular region with vertices $ (0, 0) $, $ (1, 0) $, and $ (0, 1) $. Cross-sections perpendicular to the y-axis are equilateral triangles.

$\frac{\sqrt{3}}{12}$

Applications of Integration

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Uniqlo. Emerald Triangle has three angles that are each 60 degrees because it adds upto 1 80 degrees 1 80 Divide by 360 degrees. Therefore, we know our h is sign of 60 degrees squirt of three over two times us. Which means of R A is 1/2 ass times h. Let's plug in what we know. So this is our age we're putting in right over here again. I said 1/2 s times h. Remember, the line representing the diagonal is X equals negative y plus one specified in the problem knowing that we can integrate from 0 to 1. We have squirt of three over four a square deal. Y Okay, Squirt, if there were four is a constant. Which means we're gonna be plugging this out. Pull it out on the outside of the integral from 01 We just established that we can plug end What s is negative. Why plus one squared. We established that in the previous slide, which means now we can integrate. We can do this by using the power rule. Increase the exponent by one divide by the new exponents because you can see I end up with wondered Why Cubed mice westward Plus why? Okay, now we have this next step is to plug in the bounds here. So in other words, plug in the values on the outside of the integral and we end up with squirt of three over 12 is our solution.