Find the length of the part of the line y+x+6 'inside'......

Question

Find the length of the part of the line y=x+6 'inside' (or above) the curve y=x^2. Leave answer in exact form.

Answer 1

Given y = x + 6.  Where does this line intersect the parabola y = x²?

 

The line and the parabola intersect each other when x + 6 = x²

 

Then x² - x - 6 = 0

        (x+2)(x-3) = 0

 

The points of interseciton occur when x = -2 or x = 3.

 

When x= -2, then y = x + 6 = -2 + 6 = 4 and intersection point is (-2, 4)

When x = 3, then y = x + 6 = 3 + 6 = 9 and intersection point is (3, 9)

 

Then the length of the line that intersections the parabola is the distance betwwen these two points:

 

length = √((-2 -3)² +(4 - 9)²)

          = √(25 + 25)

          = 5√(2)