Find the area of the region bounded by the parabola y = x2, the tangent line to this parabola at the point (5, 25), and the x-axis.

Answer 1

y' = 2x; at x = 5 y' = 10. Equation of tangent line: y = 25 + 10(x - 5); y = 10x - 25; x = y/10 + 2.5 and y = √x;

Area A = ∫₀²⁵(y/10 + 2.5 - √y)dy = (y²/20 + 2.5y - 2/3y^3/2)₀²⁵ = 125/4 + 62.5 - 250/3 = 125/12