Area Under a Parabola
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The figure shows a rectangle of dimensions
and
drawn under the parabola
a) Find the location
of the point associated with the largest area, and b) Find the area of the largest rectangle.
What do we know about
and
?
The point is on the given parabola, so these coordinates must satisfy the equation of the parabola.
Express this fact.
Equation 1
In order to find the maximum area, A, we need to express the area in terms of
and
?
Do that.
Equation 2
where the factor of 2 accounts for the two halves on either side of the y-axis and x1 is positive.
How can we find the maximum area?
We need to express the area in terms of one variable and differentiate.
Combine equations 1 and 2 to express the area in terms of
Differentiate.
Equation 3
Find the value of x1 which produces a maximum.
Because of the way we set up equation 2, we'll continue with the positive value.
How do we find the corresponding value of y1?
We can substitute either of these values into equation 1.
Do that.
We get
We now have determined the coordinates
required in part (a) of the problem.
How do we find the maximum area for part (b)?
We use these coordinates in equation 2. Do that.
How do we check if this is a maximum and not a minimum?
We can take the second derivative of the area and check its sign when evaluated at
. Which sign will indicate a maximum?
A negative sign is associated with a maximum, where the slope is changing toward less positive values.
Take the second derivative, using equation 3.
Do we have a maximum?
Yes, because this is negative for our positive value of x.
One more way to check is to draw a graph. Do that on your own and check with the figure below.
graph of area versus x coordinate of the corner point
To give us further insight into this result, here is a graph of the maximum rectangle under the parabola.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor