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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 9 Applications of Integration Ex 9.9

Question 1.

Find, by integration, the volume of the solid generated by revolving about the x-axis, the region enclosed by y = 2x^{2}, y = 0 and x = 1.

Solution:

Question 2.

Find, by integration, the volume of the solid generated by revolving about the x-axis, the region enclosed by y = e^{-2x} y = 0, x = 0 and x = 1.

Solution:

Question 3.

Find, by integration, the volume of the solid generated by revolving about the y-axis, the region enclosed by x^{2} = 1 + y and y = 3.

Solution:

Question 4.

The region enclosed between the graphs of y = x and y = x^{2} is denoted by R, Find the volume generated when R is rotated through 360° about x – axis.

Solution:

To find points of intersection, solving y = x^{2} and y = x, we get (0, 0) and (1, 1)

Question 5.

Find, by integration, the volume of the container which is in the shape of a right circular conical frustum as shown in the Figure.

Solution:

Question 6.

A watermelon has an ellipsoid shape which can be obtained by revolving an ellipse with major-axis 20 cm and minor-axis 10 cm about its major-axis. Find its volume using integration.

Solution:

From the given data a = 10 cm and b = 5 cm

Equation of the Ellipse

### Samacheer Kalvi 12th Maths Solutions Chapter 9 Applications of Integration Ex 9.9 Additional Questions

Question 1.

Find the volume of the solid that results when the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (a > b > 0) is revolved about the minor axis.

Solution:

Volume of the solid is obtained by revolving the right side of the curve \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) about the y-axis.

Limits for y is obtained by putting x = 0 ⇒ y^{2} = b^{2} ⇒ y = ±b.

Question 2.

Find the volume of the solid generated when the region enclosed by y = \(\sqrt{x}\), y = 2 and x = 0 is revolved about the y – axis.

Solution:

Since the solid is generated by revolving about the y-axis, rewrite y = \(\sqrt{x}\) as x = y^{2}.

Taking the limits for y, y = 0 and y = 2 (Putting x = 0 in x = y^{2}, we get y = 0)