Question

Let 𝐹 = 2𝑥𝑧𝑖 − 𝑥𝑗 + 𝑦²𝑘. Evaluate 𝑉 𝐹𝑑𝑉 is the region bounded by the surfaces 𝑥 = 0, 𝑦 = 0, 𝑧 = 𝑥² , 𝑧 = 4.

Answer 1

Here,

$F= 2xzi-xj+y^2k$

The region V is covered

(a) by keeping x and y fixed and integrating from $z=x^2 \ to\ z=4$ (base to top of column PQ),

(b) then by keeping x fixed and integrating from y=0 to y=6 (R to S in the slab)

(c) finally integrating from x=0 to x=2 (where$z=x^2$ meets z=4)

Then the required integral is :

$\int\int\int_V\ Fdv = \int_{x=0}^2\int_{y=0}^6\int_{x^2}^4(2xzi-xj+y^2k)dzdydx$

$=i\int_0^2\int_0^6\int_{x^2}^4(2xz)dzdydx-j\int_0^2\int_0^6\int_{x^2}^4(x)dzdydx+k\int_0^2\int_0^6\int_{x^2}^4(y^2)dzdydx\\\ \\=128i-24j+384k$