In this entry, we will solve the above problem using a rotational matrix.

Give this problem a try before jumping into the solution!

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Solution

By rotating the curve C and the line y = x by π/4 radians anticlockwise about the origin, the rotated volume will be given by the rotated curve about the y axis between the points at which it crosses the y axis.

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We will use the following rotational matrix

where θ = π/4.

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Using linear matrix transformation, we have the following

We have condensed the problem into finding the area in green rotating around the y-axis with the above parametric expressions.

To find the limits of the integral, we need to set x = 0 and find the corresponding t.

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Volume

Using the formula for the volume of revolution in parametric form gives us the following

The above integral is left as an exercise for readers.

We have found the final volume.