Volumes
Find the volume of the solid generated by revolv...

Question

Volumes

Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about

a. the x-axis

Accepted Answer

Welcome back everyone. Find the volume of the solid formed by revolving the region between x equals y2 and x equals 4 about the x axis, a 2 pi cubic units, b 4 pi cubic units, c 8 pi cubic units, and d 16 pi cubic units. So for this problem we're going to use the DC method formula. Volume equals pi multiplied by the integral from a to b. Of r x 2 dx, where r x is the radius of the disk, we want to find the intersection of x equals y2 and the x axis, right? The x axis is represented by y equals 0. So we basically said x equals 02, which is equal to 0. So the region extends from 0 to 4, meaning a is equal to 0 and b is equal to 4. That's how we find the limits of integration. And now we are going to express y in terms of x. If x is equal to y 2, we can take the square root of both sides and show that y is equal to plus or minus square root of x. So the region extends in y from negative square root of x to positive square root of x, and the cross section perpendicular to the x axis is a disc of radius square root of x. And hence we can apply the formula v equals integral from 0 to 4. We're going to put pi in front. And what we want to do is integrate square root of x 2 dx. That's pi integral from 0 to 4 of x dx. And finally we get pi multiplied by the integral of x is x 2 divided by 2, and we want to evaluate the result from 0 to 4. We can factor out the constant pi divided by 2 and in parentheses evaluate 42 minus 02, which gives us 16 divided by 2 multiplied by pi, that's 8 by cubic units. Which corresponds to the answer of choice c. Thank you for watching.

VolumesFind the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 abouta. the x-axis

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Volumes
Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about
a. the x-axis