Horizontal Integration - Integrating with Respect to y

Calculus 1 · Axiom Academy

LESSON Integrating with Respect to y When a region is simpler to describe sideways, slice it horizontally and integrate in y . Integrating in x sweeps vertical strips left to right; at each x the strip's height is top minus bottom. Integrating in y does the very same accumulation, only rotated: horizontal strips sweep bottom to top, and at each y the strip's width is the right boundary minus the left boundary. Vertical strips — height = top - bottom Horizontal strips — width = right - left Take the region bounded by the right-opening parabola x = y^2 and the line x = 4 . They meet where y^2 = 4 , that is at (4, 2) and (4, -2) . Watch what each slicing direction has to deal with. The parabola is the boundary, so solving x = y^2 gives two branches, and . Describing the strip means splitting the region along the x -axis into two pieces. At each y , the strip runs cleanly from x = y^2 on the left to x = 4 on the right. One expression covers the whole region, . Set it up like any area integral, but in y : the lower limit is the smallest y , the upper limit the largest, and the integrand is the strip's width, right minus left. The sweep below rises from y = -2 to y = 2 , filling the region as it goes, while the readout tracks the area gathered so far. Worked example — the region between x = y^2 and x = 4 Sketch and find the y -bounds. The curves cross at y = -2 and y = 2 , so y runs over [-2, 2] . Write x as a function of y . Right boundary ; left boundary .

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