Question

• Find the volume of the solid generated by revolving about the x-axis bounded by the curve, y2 = 9x and the line y = 3x

Answer 1

Let us find the volume of the solid generated by revolving about the x-axis bounded by the curve, $y^2 = 9x$ and the line $y = 3x.$ Firstly, let us find the points of intersection of the curve and the line: $(3x)^2=9x$ implies $9x^2=9x,$ and hence $x_1=0$ and $x_2=1.$ Let us use the formula $V=\pi\int_{x_1}^{x_2}(y_1^2(x)-y^2_2(x))dx.$

$V=\pi\int_0^1(9x-(3x)^2)dx=9\pi\int_0^1(x-x^2)dx=9\pi(\frac{x^2}{2}-\frac{x^3}{3})|_0^1= 9\pi(\frac{1}{2}-\frac{1}{3})=9\pi\cdot\frac{1}{6}=\frac{3}{2}\pi.$