ow do you find the area for a parabola that is concave up and the vertex is above the x-axis?

Question

hello in my calc class i have to prove arcimedes' formula of finding the area of a parabola and prove it is equal with intagration. My question is how do you find the area for a parabola that is concave up and the vertex is above the x-axis? i would think that it isnt possible because there is no high or base and that it doesn't exist but on all the websites i see it looks like it really is possible so i dont really know.. can you please help me!!

Answer 1

The equation for a parabola that is concave up with a vertex above the x-axis has the equation:

 

y = a(x - h)² + k where a, h and k are constants,

with a > 0 since the parabola is concave up and k > 0 since it's vertex is above the x-axis

 

if h > 0, then the parabola has its vertex in the 1st quadrant

if h < 0, then the parabola has its vertex in the 2nd quadrant

 

There are 2 possible things I can think of that the problem is asking for:

 

1. It's asking to find the area between the concave up parabola and a horizontal line y = c, where c is a constant > k (from above). In that case, you could use Arcimedes' formula.  The "height" of the parabola would be c - k and the "base" would be the difference in x-values between the 2 points where the parabola intersects the horizontal line.

 

2. It's asking for the area underneath the concave up parabola above the x-axis.  Since this is a not a concave down parabola with a < 0, you can't use Arcimedes' formula to find the area under the parabola.

 

There would need to be a range of x values defined, i.e. [x₁, x₂] where x₁ < x₂, in order to get the area under the concave up parabola, and in that case, the solution is the integral from x₁ to x₂ of a(x - h)² + k with respect to x.

 

That works out to [(a/3)(x - h)³ + kx] dx from x₂ down to x₁  

which equals (a/3)(x₂ - h)³ +kx₂ - [(a/3)(x₁ - h)³ +kx₁]

which simplifies to (a/3)[(x₂ - h)³ - (x₁ - h)³] + k(x₂ - x₁)

 

The problem as described doesn't mention a horizontal line y = c (possibility 1) or a range of x values (possibility 2) that I think are necessary for it to make sense. I would ask the teacher for clarification.