12 - Maths - Unit 7 - Integral Calculus And Its Applications

Explanation
The answer is (2) because it is the only option that is followed by a closing parenthesis. The other options are not complete and do not make sense on their own.

Explanation
The given question presents four options, and the correct answer is option (1).

Explanation
The given question provides a list of options numbered from 1 to 4. The answer is indicated as option (4), which means that the value of the expression is 4.

Explanation
The question states that the value of something is either (1), (2), (3), or (4). The answer given is (2), indicating that the value is (2).

Explanation
The area bounded by the line y = x, the x-axis, and the ordinates x = 1 and x = 2 is a triangle with a base of length 1 (from x = 1 to x = 2) and a height of 1 (since y = x). The formula for the area of a triangle is 1/2 * base * height, so the area is 1/2 * 1 * 1 = 1/2. Therefore, the correct answer is (1).

Explanation
The area between an ellipse and its auxiliary circle is equal to the difference between the areas of the ellipse and the circle. Therefore, the correct answer is (3).

Explanation
The area bounded by a parabola and its latus rectum is given by the formula A = 2/3 * a^2, where a is the length of the latus rectum. Therefore, the correct answer is (2) as it represents the formula for calculating the area bounded by a parabola and its latus rectum.

Explanation
The correct answer is (2) because when a solid is revolved about the minor axis, it forms a shape that is symmetrical along the minor axis. The volume of this solid can be calculated using the formula for the volume of a solid of revolution, which is π times the integral of the function representing the shape squared, integrated with respect to the minor axis. Therefore, the correct answer is (2) because it represents the volume of the solid obtained by revolving about the minor axis.

Explanation
When the region bounded by y = x, y = 1, x = 0 is rotated about the x-axis, it forms a solid shape known as a cone. The cone has a volume that can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height. In this case, the radius of the base is 1 and the height is also 1. Plugging these values into the formula, we get V = (1/3)π(1^2)(1) = (1/3)π. Therefore, the correct answer is (3).

Explanation
The volume of a solid obtained by revolving the area of an ellipse about its minor and major axes is in the ratio of the lengths of the axes. Since the minor axis is shorter than the major axis, the volume obtained by revolving about the minor axis will be smaller than the volume obtained by revolving about the major axis. Therefore, the correct answer is (4).

Explanation
The volume of the curve when rotated about the x-axis can be calculated using the method of cylindrical shells. This method involves integrating the circumference of the shell multiplied by its height. In this case, the curve is not specified, so it is not possible to provide a specific explanation for why the correct answer is (3).

Explanation
The volume generated by rotating a triangle about the x-axis can be found using the formula for the volume of a solid of revolution. In this case, the triangle has a base of length 3 and a height of 3. When rotated about the x-axis, it will form a cone with a radius of 3 and a height of 3. The formula for the volume of a cone is (1/3)πr^2h, where r is the radius and h is the height. Plugging in the values, we get (1/3)π(3^2)(3) = 9π. Therefore, the correct answer is (4).

Explanation
The curved surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere. In this case, the radius is given as 5. The two parallel planes are at distances 2 and 4 from the center of the sphere. Since the curved surface area is the portion of the sphere between these two planes, it is not affected by the distance of the planes from the center. Therefore, the curved surface area of the sphere is the same regardless of the distance between the planes. Hence, the correct answer is (1).

Explanation
The surface area of the solid of revolution can be found using the formula:

S = 2π∫[a,b] f(x)√(1+(f'(x))^2) dx

In this case, the region bounded by y = 2x, x = 0, and x = 2 is a triangle. The base of the triangle is 2 units and the height is 4 units.

Using the formula for the surface area of a cone, we can find the surface area of the solid of revolution:

S = πr(r + l)

where r is the radius of the base of the cone and l is the slant height.

In this case, the radius is 2 units and the slant height is 4 units.

Plugging in the values, we get:

S = π(2)(2 + 4) = 12π

Therefore, the correct answer is (1).

Explanation
The length of the arc of a curve is the distance along the curve between two given points. In this case, the length of the arc is given as 48. Therefore, the correct answer is (1).