Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by y=x and y=x2. Find V both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.
The volume of the solid obtained by rotating the region bounded by the given curves about , using the cylindrical shells and slicing.
i. By the slice method
ii. By the shell method
i. If is the radius of the typical shell, then the circumference and the height is
ii. By the shell method, the volume of the solid by rotating the region under the curve about from to is
iii. If the cross section is the disc and the radius of the disc is in terms of or , then
iv. The volume of the solid revolution about the -axis,
The region bounded by and rotated about the - axis.
As region is bounded by and rotated about the - axis.
i. By the slice method,
From the figure, as the region rotates about strip is perpendicular to -axis.
Find the intersection of the two curves by solving the simultaneous equation,
Therefore, the curves and intersects at point and
A cross section has the shape of a washer with the inner radius and the outer radius
So, find the area of cross-sectional by subtracting the area of the inner circle and the outer circle:
Therefore, the volume of the solid of revolution about -axis,
By using the fundamental theorem of calculus and the power rule of integration,
ii. By shell method,
Find the typical approximating shell with the radius as shown below.
Therefore, the circumference is and the height is
To find and , consider
So, the total volume is
The volume of the solid obtained by rotating the region bounded by the given curves is