Answer :
In the y-z plane, the curve z = 1 - y² and the curve z = y² - 1 form a bounded area around y axis. It is symmetric wrt y axis. The shape of curve is parabolic.
In the y-z plane the intersection of z = 1 - y² and z = y² - 1 gives the limits of integration and define the bounded area.
z = 1 - y² = y² - 1 => 2 y² = 2 => y = +1 or -1
So the cross section area is bounded between y = -1 and y = +1.
In the x-axis direction the solid has a uniform area. So the solid is a cylinder with the length in the x direction with cross section in the y-z plane.
Volume = cross sectional area * length
[tex]Volume = \int\limits^2_0 {Cross-section-Area} \, dx = \int\limits^2_0 ({ \int\limits^a_b {(z_2-z_1)} \, dy }) \, dx \\ \\\int\limits^2_0 ({ \int\limits^1_{-1} {(1-y^2-(y^2-1))} \, dy }) \, dx \\ \\=2\int\limits^2_0 { \int\limits^1_{-1} {(1-y^2)} \, dy } \, dx \\ \\2 \int\limits^2_0 {[y-y^3/3]_{-1}^1} \, dx =2 \int\limits^2_0 {[1-1/3-(-1+1/3)]} \, dx\\ \\=2 *\frac{4}{3} \int\limits^2_0 {1} \, dx\\ \\=2*\frac{4}{3}*[x]_0^2=2*\frac{4}{3}*2 = 16/3\\[/tex]
In the y-z plane the intersection of z = 1 - y² and z = y² - 1 gives the limits of integration and define the bounded area.
z = 1 - y² = y² - 1 => 2 y² = 2 => y = +1 or -1
So the cross section area is bounded between y = -1 and y = +1.
In the x-axis direction the solid has a uniform area. So the solid is a cylinder with the length in the x direction with cross section in the y-z plane.
Volume = cross sectional area * length
[tex]Volume = \int\limits^2_0 {Cross-section-Area} \, dx = \int\limits^2_0 ({ \int\limits^a_b {(z_2-z_1)} \, dy }) \, dx \\ \\\int\limits^2_0 ({ \int\limits^1_{-1} {(1-y^2-(y^2-1))} \, dy }) \, dx \\ \\=2\int\limits^2_0 { \int\limits^1_{-1} {(1-y^2)} \, dy } \, dx \\ \\2 \int\limits^2_0 {[y-y^3/3]_{-1}^1} \, dx =2 \int\limits^2_0 {[1-1/3-(-1+1/3)]} \, dx\\ \\=2 *\frac{4}{3} \int\limits^2_0 {1} \, dx\\ \\=2*\frac{4}{3}*[x]_0^2=2*\frac{4}{3}*2 = 16/3\\[/tex]