Mathematics

Problem

Let S be the region in the

xy

–plane bounded on the right by the curve

x=y^{2}+1

, on the left by the

y

–axis, below by the horizontal line

y=-2

, and above by

y=3

. a. Set up, but do not evaluate, an integral expression whose value equals the exact area of region S. b. Evaluate the integral from part (a) to obtain an exact numerical value for the area. c. Compute a decimal approximation of the area correct to three decimal places. d. Interpret the meaning of the area you found in part (b) in terms of accumulated horizontal "slices"; specifically, explain what each differential element represents geometrically.

Intuitive explanation

The key is to recognize this as a region where $x$ varies between two boundaries for each fixed $y$ -value—a natural setup for integration with respect to $y$ first. Since the parabola $x = y^2 + 1$ forms the right edge and the $y$ -axis ( $x = 0$ ) forms the left edge, you can "slice" horizontally across the region and let $y$ range from $-2$ to $3$ . This transforms a potentially awkward 2D shape into a straightforward iterated integral.

Intuitive explanation

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