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Let R be the region bounded by the graph of y= √x−3. the line x=4 and the x-axis. Calculate the volume of the solid generated when region R is rotated about the x-axis.

User Jimiyash
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To find the volume of a solid generated by rotating a region R around the x-axis, we use the disk method. The formula for calculating the volume is:

V = π * ∫ from a to b [f(x)]² dx

Here, f(x) = √(x - 3). This function represents the radius of the disk at any given x-value in region R. Therefore, the radius varies as we move along the x-axis from x = 4 to 3. Hence, in this case, 'a' is 4 and 'b' is 3.

So we need to find the integral of [√(x - 3)]², which is (x - 3) from 4 to 3. The anti-derivative of (x - 3) is (x² - 6x + 9)/2.

After we evaluate this antiderivative at the upper limit of integration and then at the lower limit of integration and subtract the two quantities, we obtain a result of -π/2.

This negative sign indicates that we have the limits of integration in the wrong order. So we exchange the order, from [3, 4] to [4, 3], we get π times ∫ from 4 to 3 [f(x)]² dx = -(-π/2) = π/2.

So, the volume of the solid generated when region R is rotated about the x-axis is π/2 cubic units.

User Daniel Moore
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