Question

Use a triple integral to find the volume of the given solid. The solid enclosed by the cylinder y = x^2 and the planes z = 0 and y + z = 1.

Answer 1

y = x²

z = 0 and y + z = 1

The limits of x, y and z are

x = -1 to 1

y = x² to 1

z = 0 to 1 - y

Let x² = k

The volume of the cylinder is given by

V = $\int$_-1¹ $\int$ _k ¹ $\int$ ₀^1-y dz dy dx

V = $\int$_-1¹ $\int$ _k¹  ( 1 - y ) dy dx

V = $\int$_-1¹ ( y - $\dfrac{y^2}{2}$ )_k ¹ dx

V = $\int$_-1¹ ( 1 - k - $\dfrac{1}{2}$ + $\dfrac{k^2}{2}$ ) dx

On substituting the value of k, we have

V = $\int$_-1¹ ( 1 - x² - $\dfrac{1}{2}$ + $\dfrac{x^4}{2}$ ) dx

V = $\int$_-1¹ ( $\dfrac{1}{2}$ - x² + $\dfrac{x^4}{2}$ ) dx

V = ( $\dfrac{1}{2}$x - $\dfrac{x^3}{3}$ + $\dfrac{x^5}{10}$ )_-1 ¹

V = [ $\dfrac{1}{2}$(1 + 1) - $\dfrac{(1+1)^3}{3}$ + $\dfrac{(1+1)^5}{10}$ ]

V = [ 1 - $\dfrac{8}{3}$ + $\dfrac{32}{10}$ ]

V = 1.533 cubic units.