Exercise 8.1(Application of Integrals)
Chapter 8 Exercise 8.1 ncert math solution class 12
Question1: Find the area of the region bounded by the curve and the lines
and the x-axis.
Solution : The area of the region bounded by the curve, , the lines, x = 1 and x = 4, and the X-axis is the area E B C F.
Area of
units
Question 2: Find the area of the region bounded by and the x – axis in the first quadrant.
Solution : The area of the region bounded by the curve, , and x = 4, and the x-axis, is the area EBCF.
Area of
units
Question 3: Find the area of the region bounded by and the y-axis in the first quadrant.
Solution : The area of the region bounded by the curve, , and y = 4, and the y-axis is the area
.
Area of
units
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 Area of OAB
Area of
Therefore, area bounded by the ellipse =4 × Area of 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 of OAB
Therefore, area bounded by the ellipse Area of
units
Question 6: Find the area of the region in the first quadrant enclosed by x-axis, line and the
Solution : The area of the region bounded by the circle, ,line
and the x-axis is the area OAB.
Solving and
and
The point of intersection of the line and the circle in the first quadrant is .
Area Area
Area
Area of
….(i)
…..(ii)
Adding the equation (1) and (2), we get
The total area of units
Therefore, area enclosed by x – axis, the line and the circle
in the first quadrant is
square units.
Question 7: Find the area of the smaller part of the circle cut off by the line
.
Solution : It can be observed that the area is symmetrical about x -axis.
Given,
Area of ABC
Therefore, the area of smaller part of the circle, , cut off by the line
is
square units.
Question 8: The area between and
is divided into two equal parts by the line
, 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 Area
It can be observed that the given area is symmetrical about x-axis.
⇒ Area Area
Area
…..(i)
Area of
…..(ii)
From (1) and (2), we have
Therefore, the value of a is .
Question 9: Find the area of the region bounded by the parabola and
Solution: The area bounded by the parabola, , and the line,
, can be represented as The given area is symmetrical about y-axis.
⇒ Area OACO = Area ODBO
Solving and y = x
and y = 1 and y = 0
The point of intersection of parabola, , and line, y = x, is A(1, 1) and (0, 0).
Area of Area
Area of
Area of
⇒ Area of Area of
Therefore, required area units
Question 10: Find the area bounded by the curve and the line
.
Solution : The area bounded by the curve and the line
is represented by the shaded area OBAO.
Solving and
and
aand y=1 and
Let A and B be the points of intersection of the line and parabola. Coordinates of point A are and point B are (2, 1).
We draw AL and BM perpendicular to x-axis.
It can be observed that,
Area OBAO
Question 11:Find the area of the region bounded by the curve and the line x=3.
Solution : The region bounded by the parabola, , and the line, x=3, is the area OABO.
The area OABO is symmetrical about x-axis.
Area of (Area of
Area
Therefore, the required area is units
Question 12: Area lying in the first quadrant and bounded by the circle and the lines x = 0 and x = 2 is
(A)
(B)
(C)
(D)
Solution : The correct answer is (A).
The area bounded by the circle and the lines, x = 0 and x = 2, in the first quadrant is represented as
Area of
units
Thus, the correct answer is (A).
Question 13: Area of the region bounded by the curve -axis and the line y = 3 is
(A) 2
(B)
(C)
(D)
Solution : The correct answer is (B)
The area bounded by the curve, -axis and y = 3 is represented as
Area
units
Thus, the correct answer is (B).
Chapter 8: Application of Integrals Class 12
Exercise 8.1 ncert math solution class 12
Exercise 8.2 ncert math solution class 12