Find the area of the region bounded by the parabola y 2 = 4ax and its latus rectum x = a (full region, both above and below the x-axis).

Find the area of the region bounded by…

Answer: (8/3)a 2.

A (8/3)a2
B (16/3)a2
C (4/3)a2
D 4a2

Correct answer: A. (8/3)a2

Explanation: By symmetry, total area = 2 times the upper-half area. Upper-half area = integral from 0 to a of 2sqrt(ax) dx = 2sqrt(a)[(2/3)x3/2] from 0 to a = 2sqrt(a)(2/3)a3/2 = (4/3)a2. Doubling for both halves gives 2 times (4/3)a2 = (8/3)a2.

The definite integral ∫ₐᵇf(x)dx computes the exact area of the shaded region bounded by the curve, the x-axis, and the vertical lines x=a and x=b - the same idea behind the Riemann sum, but evaluated exactly rather than approximated by rectangles.

Concept context

Using definite integrals to compute the area under a curve, the area between two curves, and the areas enclosed by standard curves like circles, parabolas, and ellipses.

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Find the area of the region bounded by the parabola y<sup>2</sup> = 4ax and its latus rectum x = a (full region, both above and below the x-axis).

Concept context

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