Integrating Over a Plane Region--X

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

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This animation shows how horizontal 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 horizontal line segments, which corresponds to choosing to integrate first in the horizontal coordinate. Note that the horizontal line segments that sweep out this region are of two different kinds: Some begin on the parabola and end on the line, while others begin on the parabola and end elsewhere on the parabola. Compare this animation with Integrating Over a Region in the Plane: y-axis First. (11/20/07)