The geometric meaning of double integral is the volume of a cylinder with a curved top. Now the curved top is a conical surface, so the volume of the curved top cylinder is obtained. In fact, the volume of a cone is 2/3SR obtained by subtracting the volume of a cylinder with the projection as the base and R as the height.

From -R to r, the upper and lower parts of the sphere are symmetrical, so the value range of t should be 0 to r, and the final integration result should be multiplied by 4.

∫(dao0, π)da∫(0, r) radical (r? -t？ )×(- 1/2)d(R？ -t？ )

=∫(0，π)da (- 1/2)(2/3)(R？ -t？ Power of 3/2) from 0 to R.

Cubic da =∫(0, π) 1/3R

= 1/3πR cube

V=4× 1/3πR cube =4/3πR cube.

Extended data:

When the integrand function is greater than zero, the double integral is the volume of the cylinder.

When the integrand function is less than zero, the double integral is the negative value of the cylinder volume. ?

In the spatial rectangular coordinate system, the double integral is the algebraic sum of the cylindrical volumes of all parts in the region, which is positive above the xoy plane and negative below the xoy plane. The volume formula of the curved cylinder surrounded by the curved surface of some special integrand f(x, y) and the bottom surface of d is known, which can be calculated by the geometric meaning of double integral.

Baidu Encyclopedia-Double Integral