Question
Find the area of the region bounded by the parabola y = 5x2, the tangent line to this parabola at (5, 125), and the x-axis.
its not 625/3
its not 625/3
Answers
oobleck
the tangent has slope y' = 10x
so at (5,125) the tangent line is
y-125 = 50(x-5)
y = 50x - 125
The boundary changes where the line intersects the x-axis, so the area is thus seen to be
∫[0,5/2] 5x^2 dx + ∫[5/2,5] (5x^2 - (50x-125)) dx = 625/24 + 625/24 = 625/12
or, using horizontal strips of width dy,
∫[0,125] ((y+125)/50 - √(y/5)) dy = 625/12
so at (5,125) the tangent line is
y-125 = 50(x-5)
y = 50x - 125
The boundary changes where the line intersects the x-axis, so the area is thus seen to be
∫[0,5/2] 5x^2 dx + ∫[5/2,5] (5x^2 - (50x-125)) dx = 625/24 + 625/24 = 625/12
or, using horizontal strips of width dy,
∫[0,125] ((y+125)/50 - √(y/5)) dy = 625/12