Question

Use the shell method to write and evaluate the definite integral that represents the volume of the solid \[ y=\sqrt{x} \]

Answer 1

Explanation:

To find the volume of the solid generated by rotating the curve y=xy=x

​ about the y-axis using the shell method, you need to integrate with respect to y and consider infinitesimally thin vertical shells. The formula for the volume of a solid using the shell method is:

V=2π∫abr(x)h(x)dxV=2π∫ab​r(x)h(x)dx

where:

aa and bb are the limits of integration along the y-axis (in this case, the y-values that correspond to the region of interest).

r(x)r(x) is the radius of the shell (the distance from the axis of rotation to the outer edge of the shell).

h(x)h(x) is the height of the shell (the infinitesimal thickness of the shell).

In this case, we'll integrate with respect to yy, so we need to express r(x)r(x) and h(x)h(x) in terms of yy. First, let's find the limits of integration. The curve y=xy=x

​ starts at x=0x=0 and ends at x=bx=b. To find bb, set y=xy=x

​ equal to a constant, which represents the upper limit of integration:

x=c ⟹ x=c2x

​=c⟹x=c2

So, b=c2b=c2.

Now, we'll express r(x)r(x) and h(x)h(x) in terms of yy:

r(x)r(x): The distance from the axis of rotation (y-axis) to a point on the curve y=xy=x

​ is simply xx, so r(x)=xr(x)=x.

h(x)h(x): The thickness of the shell along the y-axis is dydy (infinitesimally small change in y). Therefore, h(x)=dyh(x)=dy.

Now, we have everything we need to set up and evaluate the definite integral for the volume:

V=2π∫0cx dyV=2π∫0c​xdy

We already found that b=c2b=c2, so the limits of integration are from 0 to cc.

Integrate with respect to yy:

V=2π∫0cx dy=2π∫0cx dy=2π∫0cx dy=2π∫0cy2 dyV=2π∫0c​xdy=2π∫0c​xdy=2π∫0c​xdy=2π∫0c​y2dy

Now, integrate:

V=2π[13y3]0c=2π(13c3−13(0)3)=23πc3V=2π[31​y3]0c​=2π(31​c3−31​(0)3)=32​πc3

So, the volume of the solid generated by rotating the curve y=xy=x

​ about the y-axis is 23πc332​πc3, where cc is the value where the curve intersects the y-axis (the upper limit of integration).