The area of d and a is the area of the trapezoid of the curve (p/6) by subtracting the tangent from the integral of the curve px^3 in [0, 1], and the x axis, x = 1.
S=4∫∫(xy/a)dxdy
=(4/a)[∫(0->； π/2)dθ][∫(0-& gt； a)(r^3sinθcosθ)dr
=a^3/2
Extended data:
General theorem
Theorem 1: If f(x) is continuous in the interval [a, b], then f(x) is integrable in [a, b].
Theorem 2: If the interval f(x) is bounded on [a, b] and there are only finite discontinuous points, then f(x) is integrable on [a, b].
Theorem 3: Let f(x) be monotone in the interval [a, b], then f(x) can be integrated in [a, b].
Definite integral and indefinite integral seem to have nothing to do, but they are closely related in essence because of the support of a mathematically important theory. It seems impossible to subdivide a graph infinitely and then accumulate it, but because of this theory, it can be transformed into calculating integral.
Baidu encyclopedia-definite integral