What is the volume of the solid formed by the line?

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What is the volume of the solid formed by the line?

The graph of $y = 2 sin (\frac{π x}{2})$ $y=2sin((pix)/2)$ from x = 0 to x = 2 is rotated around the y-axis to form a solid figure. Find the volume of the solid.

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Answer 1 · 2018-05-24T17:44:43

$V = π \int_{0}^{2} {(2 sin (\frac{π x}{2}))}^{2} \cdot d x = 4 π$ $color(blue)[V=piint_0^2(2sin((pix)/2))^2*dx=4pi]$

Explanation:

we will use Disk method to calculate the volume of the solid generated from rotating the curve around $x-axis$ $"x-axis"$

the interval of the integral of volume $x \in [0, 2]$ $x in [0,2]$

The volume of the solid between the curve and the axis is given by:

$V = π \int_{a}^{b} y^{2} \cdot d x$ $V=piint_a^by^2*dx$

$V = π \int_{0}^{2} {(2 sin (\frac{π x}{2}))}^{2} \cdot d x$ $V=piint_0^2(2sin((pix)/2))^2*dx$

$V = π \int_{0}^{2} (4 {sin}^{2} (\frac{π x}{2})) \cdot d x$ $V=piint_0^2(4sin^2((pix)/2))*dx$

$V = 4 π \int_{0}^{2} (\frac{1}{2} - \frac{1}{2} \cdot cos (π x)) \cdot d x$ $V=4piint_0^2(1/2-1/2*cos(pix))*dx$

$V = 2 π \int_{0}^{2} (1 - cos (π x)) \cdot d x$ $V=2piint_0^2(1-cos(pix))*dx$

$= 2 π {[x - \frac{1}{π} sin (π x)]}_{0}^{2} = [(2 - \frac{1}{π} \cdot sin 2 π) - (- \frac{1}{π} sin 0)]$ $=2pi[x-1/pisin(pix)]_0^2=[(2-1/pi*sin2pi)-(-1/pisin0)]$

$2 - 0 + 0 = 4 π$ $2-0+0=4pi$

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What is the volume of the solid formed by the line?

The graph of $y = 2 sin (\frac{π x}{2})$ $y=2sin((pix)/2)$ from x = 0 to x = 2 is rotated around the y-axis to form a solid figure. Find the volume of the solid.

1 Answer

Explanation:

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What is the volume of the solid formed by the line?

The graph of y=2sin(πx2)y=2sin((pix)/2) from x = 0 to x = 2 is rotated around the y-axis to form a solid figure. Find the volume of the solid.

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Explanation:

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The graph of $y = 2 sin (\frac{π x}{2})$ $y=2sin((pix)/2)$ from x = 0 to x = 2 is rotated around the y-axis to form a solid figure. Find the volume of the solid.