Find the volume of a pyramid with height h and rectangular base with dimensions b and 2b.

firestopper53sh

firestopper53sh

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2022-08-19

Finding volume of the solid of revolution
The function is:
y = sin ( x 2 ) and the boundaries are y = 0, x = 0, and x = π .
The function is rotated around the y-axis, and since the functions are difficult to express in terms of y and integrate with respect to dy, I use the formula of V = 2 π 0 π sin ( x 2 ) d x
My two questions here are:
1) What method can I use to solve the integral of sin ( x 2 )?
2) How do I limit the boundary in some way to not include the area below the x-axis?

Answer & Explanation

Lamar Casey

Lamar Casey

Beginner2022-08-20Added 8 answers

Step 1
V = 0 π ( 2 π x ) sin ( x 2 ) d x where 2 π x is the radius of the shell and sin ( x 2 ) is the height of the shell. Unlike your previous integral, this integral has a closed form, and has a straightforward solution by substitution.
So as for your first question, the integral of sin 2 ( x ) is not expressible in closed form. You could write it as π 2 S ( 2 ), where S(x) is the Fresnel S Integral.
Step 2
Volume of revolution around the y axis using shells is 2 π 0 a x f ( x ) d x .
Regarding your second question sin x 2 0 when 0 x 2 π . Thus we don't need to worry about any area below the x axis.

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