Calculus Problem

<p>So, I am self studying calculus, and ran across a problem that i have no idea how to solve.
This is barron's AP calculus, 9th Edition, Chapter 7 : Applications of Integration to Geometry, Problem # 26.</p>

<p>The base of a solid is the region bounded by the parabola x^2 = 8y, and the line y = 4, and each plane section perpendicular to the y axis is an equilateral triangle. The volume of the solid is _____</p>

<p>The book says that the answer is 64 rad(3), but i have no idea how it got there.</p>

<p>I understand that i have to integrate the area * dx then multiply it by 2. But i have no idea how to get the area.</p>

<p>Can any one help me with this?? Thank you! (Please show full work)</p>

<p>I'm not going to show full work, because that is not allowed here, plus I wouldn't be helping you if I just spoonfed you with the answer.</p>

<p>I'm assuming you know how to find the area of any triangle. So given an equilateral triangle, how do you find the height?</p>

<p>Think right triangles. After that the rest is easy.</p>

<p>Draw a sketch of the parabola and the line y=4. Now try to envision the equilateral triangle at a particular y-value, and you'll see that the base is 2x and the height is x<em>rad3. Thus the area of each triangle above a certain y-value is x^2</em>rad3. You then need to integrate x^2<em>rad3</em>dy from y=0 to y=4. Since it is given that x^2=8y, you can rewrite this as 8y<em>rad3</em>dy. Integrating, you have 4y^2*rad3 from y=0 to y=4. This gives you 64rad3. If any step isn't clear, feel free to let me know.</p>

<p>i do not understand how the base is 2x :(
im sorry for being clueless if this is a stupid question/</p>

<p>Look at your graph. What does x, any arbitrary x, represent?</p>

<p>Now look again, what does 2x represent?</p>

<p>The graph contains the answers to your questions. Make a quick sketch of the parabola, the line y=4, and then the equilateral triangle(s).</p>

<p>x = distance from origin to whereever.</p>

<p>2x = double the distance?</p>

<p>if i plug in values, y = 1/2x.. not 2x.</p>

<p>Am i doing something wrong?</p>

<p>Oh never mind. I get it now!!!</p>

<p>Thank you!</p>

<p>Isn't it a bit early to self-study now?</p>

<p>Well at least he/she knows what to work on...</p>

<p>is it too early to self study now?? 0.0??</p>

<p>AP test isn't until May?</p>

<p>I'm not in calc yet, but I actually understood what was done! U guys split the triangle in 1/2 resulting in a 30-60-90 triangle. The base of this triangle is x, the height x<em>rad(3), and the hypotnuse 2x. Then the base of the 30-60-90 is 1/2 the base of the equilateral triangle, so the entire triangle has base 2x and height x</em>rad(3). Then u did the integration part, which is what I have to look forward to learning next year :)</p>

<p>^ good job.</p>

<p>goood job!
ur better than me at this! :D</p>

<p>Looks like from his screen name he goes to TJ. Volumes of solids will be the least of his worries in the future. ;)</p>

<p>LOL thank you.</p>

<p>But CC really overrates TJ.</p>

<p>I wish I went to TJ.</p>

<p>Hahaha where do you go?</p>

<p>I'm from California.</p>

<p>Is TJ full of science and math nerds? lol</p>

<p>If that is the same TJHS that competed at MAO nationals last week, then it is really good at math.</p>