The discussion focuses on calculating the volume of cross sections formed by isosceles right triangles perpendicular to the x-axis, bounded by the curve y=x² and the line x=3. The correct volume formula derived from the area of the triangles is V=(1/2)x⁴Δx, which leads to the integral V=x⁵/10 evaluated from 0 to 3, resulting in a final volume of 243/10. Participants clarify that the leg of the triangle lies perpendicular to the x-axis, and emphasize the importance of using the given variables without introducing new ones.
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HallsofIvy said:Are we to assume that the cross sections perpendicular to the x-axis are "isosceles right triangles"- with one leg perpendicular to the x-axis or the hypotenuse?
I think what you are trying to do is correct but your notation is very strange. You are given information about x and y- use them, not new variables! Assuming it is the leg that lies perpendicular to the x-axis then the other leg is just y= x^2. The area of such a triangle is (1/2)(x^2)(x^2)= x^4/2. The "thickness" of each such cross-section is \Delta x so the volume is (1/2)x^4\Delta x. In the limit that becomes an integral.