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The area of the region bounded by the curve,
y
2
=
x
, the lines,
x
= 1 and
x
= 4, and the
x
-axis is the area ABCD.
Question 3.
Find the area of the region bounded by
x
2
= 4
y
,
y
= 2,
y
= 4 and the
y
-axis in the first quadrant.
Solution :
The area of the region bounded by the curve,
x
2
= 4
y
,
y
= 2, and
y
= 4, and the
y
-axis is the area ABCD.
Question 4. Find the area of the region bounded by the ellipse
Solution :
The given equation of the ellipse,
It can be observed that the ellipse is symmetrical about
x
-axis and
y
-axis.
∴ Area bounded by ellipse = 4 × Area of OAB
Therefore, area bounded by the ellipse = 4 × 3π = 12π units
Question 5. Find the area of the region bounded by the ellipse
Solution :
The given equation of the ellipse can be represented as
It can be observed that the ellipse is symmetrical about
x
-axis and
y
-axis.
∴ Area bounded by ellipse = 4 × Area OAB
Therefore, area bounded by the ellipse = 4 x3π/2 = 6π units.
Question 6.
Find the area of the region in the first quadrant enclosed by
x
-axis, line x = √3y and the circle x
2
+ y
2
= 4
Solution :
The area of the region bounded by the circle, x
2
+ y
2
= 4,x = √3y and the
x
-axis is the area OAB.
The point of intersection of the line and the circle in the first quadrant is (√3,1).
Area OAB = Area ΔOCA + Area ACB
Question 7. Find the area of the smaller part of the circle
x
2
+
y
2
=
a
2
cut off by the line x = a/√2
Solution :
The area of the smaller part of the circle,
x
2
+
y
2
=
a
2
, cut off by the line, x = a/√2, is the area ABCDA.
It can be observed that the area ABCD is symmetrical about
x
-axis.
∴ Area ABCD = 2 × Area ABC
Therefore, the area of smaller part of the circle,
x
2
+
y
2
=
a
2
, cut off by the line,
Question
8. The area between
x
=
y
2
and
x
= 4 is divided into two equal parts by the line
x
=
a
, find the value of
a
.
Solution :
The line,
x
=
a
, divides the area bounded by the parabola and
x
= 4 into two equal parts.
∴ Area OAD = Area ABCD
It can be observed that the given area is symmetrical about
x
-axis.
⇒ Area OED = Area EFCD
Question
9. Find the area of the region bounded by the parabola
y
=
x
2
and y = |x|
Solution :
The area bounded by the parabola,
x
2
=
y
,and the line,y = |x|, can be represented as
The given area is symmetrical about
y
-axis.
∴ Area OACO = Area ODBO
The point of intersection of parabola,
x
2
=
y
, and line,
y
=
x
, is A (1, 1).
Area of OACO = Area ΔOAM – Area OMACO
Therefore, required area = 2[1/6] = 1/3 units
Question
10. Find the area bounded by the curve
x
2
= 4
y
and the line
x
= 4
y
– 2
Solution :
The area bounded by the curve,
x
2
= 4
y
, and line,
x
= 4
y
– 2, is represented by the shaded area OBAO.
Let A and B be the points of intersection of the line and parabola.
Coordinates of point A are (-1, 1/4).
Coordinates of point B are (2, 1).
We draw AL and BM perpendicular to
x
-axis.
It can be observed that,
Area OBAO = Area OBCO + Area OACO … (1)
Then, Area OBCO = Area OMBC – Area OMBO
Similarly, Area OACO = Area OLAC – Area OLAO
Therefore, required area =
units
Question
11.
Find the area of the region bounded by the curve
y
2
= 4
x
and the line
x
= 3
Solution :
The region bounded by the parabola,
y
2
= 4
x
, and the line,
x
= 3, is the area OACO.
The area OACO is symmetrical about
x
-axis.
∴ Area of OACO = 2 (Area of OAB)
Therefore, the required area is 8√3 units.
Question
12. Choose the correct answer:
Therefore, option (A) is correct.
Question
13. Choose the correct answer:
Therefore, option (B) is correct.
