SOLUTION: find the area bounded by the curve y^2=9x and its latus rectum

Question 1110815: find the area bounded by the curve y^2=9x and its latus rectum
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!

In this case, the latus rectum is part of the line .
Parabola, focus, directrix and latus rectum look like this
The area we want is the area between the red/green curve and the blue latus rectum.

ONE WAY:
As the halves above and below the x-axis are symmetrical, we can calculate that area as twice the area above the x-axis.
When we only consider that half, with ,
is equivalent to , so

As the antiderivative is ,
that area is
.

ALTERNATIVELY,
you could do like you may have to do in other cases and interchange variables
(or take x as a function of y).
is equivalent to taking x as a function of y.
That curve and intersect at .
In other words, the endpoints of the latus rectum are the points with
.
We can calculate the area between the functions of ,
and ,
in the interval from to as
,
or better yet as
. As ,
the area would be calculated as

.
RELATED QUESTIONS
find the area bounded by the curve x^2=8y and its latus... (answered by math_helper)

find the area bounded by the curve x^2=8y and its latus rectum (answered by htmentor)

Find the area bounded by y-axis , the curve x^2=4a(y-2a) and... (answered by rothauserc)

find the area bounded by the curve 9x^2 + 25y^2 + 18x - 100y =... (answered by ikleyn)

Find the area bounded by the x-axis, the curve y = e2x and the ordinates x = 2 and x = 3. (answered by Earlsdon)

To find the area bounded by the curve of the graph 9-x^2,,and to find the volume bounded... (answered by Alan3354)

Find the latus rectum, vertex and directrix of the parabola... (answered by MathLover1,josgarithmetic,MathTherapy)

Find the area bounded by the curve... (answered by robertb)