please derive the area and volume of a pyramid using calculus?

Answer 1

How many sides on the pyramid? 3, 4,...? Is it regular or irregular? Those things are important, because you can't give a calculus formula for a general pyramid except V = Bh, where B is the area of the base and h is the height, and A = B + P where P is the sum of areas of the pyramid sides

Answer 2

Volume is integrated in this way: Let A = area of the base of the pyramid and h = height of the pyramid, and x = distance from the apex. Then we know that A(x) = A (x/h)^2. We integrate this with respect to x:

S(0 to h) A * x^2 / h^2 dx = S(0 to h) A * 1/3 x^3 / h^2 = 1/3 A h

Area is integrated in this way: Let P be the vector location of the apex, and Q be the vector location of any point on the base. Let B(s) be the vector function of the perimeter of the base, where s is the length parameter of the length, S being the total perimeter length. We integrate the side and base of the pyramid as follows with respect to s:

S(0 to S) Sqrt((P-B(s))o(P-B(s))) ds + S(0 to S) Sqrt((Q-B(s))o(Q-B(s))) ds

where the symbol o denotes the vector dot product. Good luck with this one.