Integrating to find bounded area
PART 2: with respect to y
We have been determining area
bounded by two curves by writing
and evaluating a definite integral.
We have had area bounded on the
top and the bottom by two functions
and used the formula:
right boun dary
∫
(top curve − bottom curve) dx
left boun dary
This type of integration
is done with respect to x.
B
2x
2
∫A [ 3 − (x − 2x − 1)] dx
But…….some area is bounded on
the right and left
When this happens we follow the
pattern:
top boun dary
∫
(right curve − left curve) dy
bottom boun dary
This integration
With respect to y
The integral for this bounded area
is:
1
∫ [y + 2 − (1− y
2
)] dy
−1
NOTICE
EVERYTHING
IS IN TERMS
OF Y
This integration
With respect to y
top boun dary
∫
bottom boun dary
(right curve − left curve) dy
Try another example: the area
bounded by x=y2-4 and y=2-x
First, determine the points of
intersection………
x=y2-4, x=2-y
y2-4=2-y
y2+y-6=0
(y+3)(y-2)=0
y=-3, y=2
Then write the integral……
x=y2-4, x=2-y
y2-4=2-y
y2+y-6=0
(y+3)(y-2)=0
y=-3, y=2
2
∫ [2 − y − (y
−3
2
− 4)] dy
One more……..determine the area
bounded by x=y2+1, y=2, y=1, x=0
First solve x=y2+1
for y:
Then sketch the graph:
Finally, write the integral
2
∫ [y
1
2
+ 1] dy
x-1=y2,
y = ± x −1