Volume with Square Cross Sections

Calculus 2 · Axiom Academy

EXAMPLE Volume with Square Cross Sections Find the volume of a solid whose cross sections perpendicular to the x-axis are squares. Base region: the area between the curves y = x and y = x^2 . Cross sections: squares taken perpendicular to the x-axis. Find the volume of the resulting solid. Each vertical slice of the base region is the side of a square that stands perpendicular to the x-axis. Nice work! You found the volume using the cross-sectional area method. Here is what each step did: Find the integration bounds: set the curves equal to locate where they intersect — here x = 0 and x = 1 . Identify the side length: for square cross sections, the side equals the distance between the curves, (upper) − (lower) = x - x^2 . Build the cross-sectional area: square the side length to get A(x) = (x - x^2)^2 , which varies with position x . Set up the integral: sums every infinitesimal cross-sectional area across the region. Expand and integrate: expand before integrating, then apply the power rule term by term to reach . This method extends to other cross-sectional shapes (rectangles, triangles, semicircles) by swapping in the right area formula. The recipe is always the same: find the side length(s), build the area, then integrate.

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