Calculus

Volume of Revolution

Volume of Revolution - Problem Solving

         

If the surface area of the hypersphere (sphere in 4 dimensions) is 2π2R3 2 \pi ^2 R^3 , what would be the volume of the hypersphere?

Consider a three dimensional object whose base is a semicircle with diameter AB=28AB = 28. If each cross section of the three dimensional object cut by a plane perpendicular to AB\overline{AB} is a semicircle, as shown in the diagram above, what is the volume of the three dimensional object?

Consider a three-dimensional shape standing on the floor with height π2 cm\frac{\pi}{2}\text{ cm} such that the area of a cross section parallel to the floor is 14sinxcosx cm214\sin x\cdot \cos x \text{ cm}^2 for any height 0xπ2 cm.0 \le x \le \frac{\pi}{2}\text{ cm}. What is the volume of the three-dimensional shape in cm3\text{cm}^3?

Consider a three-dimensional shape standing on the floor with height 17 cm17 \text{ cm} such that the area of a cross section parallel to the floor is 3217x cm2 \frac{3}{2} \sqrt{17-x} \text{ cm}^2 for any height 0x17.0 \le x \le 17 . What is the volume of the three-dimensional shape in cm3\text{cm}^3?

In the above diagram, the equation of the curve is y=ax=2xy=\sqrt{ax}=\sqrt{2x} and the equation of the line is y=mx.y=mx. Let VxV_x and VyV_y be the volumes of the solids obtained by revolving the shaded region bounded by the curve and line about the xx-axis and yy-axis, respectively. If Vx=Vy,V_x=V_y, what is the value of m?m?

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