Integrating Over a Plane Region--Y

Integrating Over a Plane Region: Along the y-axis First

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This animation shows how vertical line segments sweep out a certain plane region. (See notes below.)

When we set up an iterated integral in cartesian coordinates, we must decide whether to carry out the first integration in the horizontal coordinate or in the vertical coordinate. This animation shows how the plane region bounded by the curve y = x + 1 and the curve y = 1 + 4 x – x² is swept out by vertical line segments. This is how we must think about generating the area if we choose to integrate first in the vertical coordinate. Note that the vertical line segments that sweep out this region are all of one kind: They begin on the line and end on the parabola. Compare this animation with Integrating Over a Region in the Plane: x-axis First. (11/20/07)