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Find the area of the region bounded by x + 1 = 0, y = 0, y = x² + x + 1, and the tangent to y = x² + x + 1 at x = 1. If this area is k, find the value of 3k.

1. 1
2. 2
3. 3
4. 4

Correct answer: 4

Solution

Tangent at x=1 is y=3x. Boundaries: x=-1 (left), y=0 (bottom), parabola y=x²+x+1 (top/right), tangent y=3x. The enclosed region needs careful identification. The parabola and tangent meet at x=1 (y=3). Integrating the area between these curves from x=-1 to x=1 and accounting for y=0 gives k=4/3, so 3k=4.

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