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Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE PDF Download

Q1: Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is.
Ans: 
Given,
Equation of circle is x2+ y2 = 4……….(i)
x2+ y2 = 22
y = √(22 – x2) …………(ii)
Equation of a lines is x + y = 2 ………(iii)
y = 2 – x
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE

Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE

Therefore, the graph of equation (iii) is the straight line joining the points (0, 2) and (2, 0).
From the graph of a circle (i) and straight-line (iii), it is clear that points of intersections of circle
(i) and the straight line (iii) is A (2, 0) and B (0.2).
Area of OACB, bounded by the circle and the coordinate axes is
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
= [ 1 × √0 + 2 sin-1(1) – 0√4 – 2 × 0]
= 2 sin-1(1)
= 2 × π/2
= π sq. units
Area of triangle OAB, bounded by the straight line and the coordinate axes is
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
= 4 – 2 – 0 + 0
= 2 sq.units
Hence, the required area = Area of OACB – Area of triangle OAB
= (π – 2) sq.units

Q2: Find the area of the curve y = sin x between 0 and π.
Ans: 
Given,
y = sin x
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
Area of OAB
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
= – [cos π – cos 0]
= -(-1 -1)
= 2 sq. units

Q3: Find the area enclosed by the ellipse x2/a2 + y2/b2 =1.
Ans:
Given,
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
We know that,
Ellipse is symmetrical about both x-axis and y-axis.
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
Area of ellipse = 4 × Area of AOB
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
Substituting the positive value of y in the above expression since OAB lies in the first quadrant.
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
= 2ab × sin-1(1)
= 2ab × π/2
= πab
Hence, the required area is πab sq.units.

Q4: Find the area of the region bounded by the two parabolas y = x2 and y= x.
Ans: 
Given two parabolas are y = x2 and y2 = x.
The point of intersection of these two parabolas is O (0, 0) and A (1, 1) as shown in the below figure.
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
Now,
y2 = x
y = √x = f(x)
y = x2 = g(x), where, f (x) ≥ g (x) in [0, 1].
Area of the shaded region

Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
= (⅔) – (⅓)
= ⅓
Hence, the required area is ⅓ sq. units.

Q3: Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant.
Ans: 
We can draw the figure of y2 = 9x; x = 2, x = 4 and the x-axis in the first curve as below.
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
y= 9x
y = ±√(9x)
y = ±3√x
We can consider the positive value of y since the required area is in the first quadrant.
The required area is the shaded region enclosed by ABCD.
Important Questions: Applications of Integrals | Mathematics (Maths) Class 12 - JEE
= 2 [(2)3 – (√2)3]
= 2[8 – 2√2]
= 16 – 4√2
Hence, the required area is 16 – 4√2 sq.units.

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FAQs on Important Questions: Applications of Integrals - Mathematics (Maths) Class 12 - JEE

1. What are some real-life applications of integrals?
Ans. Integrals have numerous real-life applications. Some common examples include finding the area under a curve, calculating the total distance traveled by an object, determining the volume of irregular shapes, analyzing population growth, and solving problems related to physics and engineering.
2. How do integrals help in calculating areas?
Ans. Integrals help in calculating areas by summing up infinitesimally small rectangles under a curve. By dividing the region into narrow strips and finding the area of each strip, we can then add up all these areas using integration to obtain the total area under the curve.
3. Can integrals be used to find the volume of irregular objects?
Ans. Yes, integrals can be used to find the volume of irregular objects. By slicing the object into infinitesimally thin slices, calculating the volume of each slice, and then integrating these volumes, we can determine the total volume of the irregular object.
4. How are integrals applied in physics?
Ans. Integrals are extensively used in physics to solve a wide range of problems. They help in calculating quantities such as displacement, velocity, acceleration, work, energy, and electric/magnetic fields. Integrals are particularly useful in analyzing motion, forces, and systems governed by differential equations.
5. Are there any practical applications of integrals in engineering?
Ans. Yes, integrals have numerous practical applications in engineering. They are used to analyze and design structures, calculate moments of inertia, determine fluid flow rates and pressure distributions, solve differential equations in electrical circuits, optimize system performance, and model various physical phenomena encountered in engineering disciplines.
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