Question

For the solid S described, do the following:

Answer 1

Answer:

(a) and (b) see pictures attached

(c) V = 16/35

Step-by-step explanation:

(a) Sketch the base of S in the xy-plane.

See picture 1 attached

(b) Sketch a three-dimensional picture of S with the xy-plane as the floor.

See picture 2 attached

(c) Compute the volume of S.

The volume is given by the triple integral

$\displaystyle\iiint_{S}zdzdydx$

The cross-sections perpendicular to the x-axis are squares so  

$z=1-x^2$

The region S is given by the following inequalities

$0\leq x\leq 1\\\\x^2\leq y\leq 1\\\\0\leq z\leq 1-x^2$

Therefore

$\displaystyle\iiint_{S}zdzdydx=\displaystyle\int_{0}^{1} \displaystyle\int_{x^2}^{1} \displaystyle\int_{0}^{1-x^2} (1-x^2)dzdydx=\\\\\displaystyle\int_{0}^{1}(1-x^2)(1-x^2)(1-x^2)dx=\displaystyle\int_{0}^{1}(1-x^2)^3dx=\displaystyle\frac{16}{35}$

So the volume V of the solid S is

V=16/35