Question

örel thu Relloving rion bundel by the funcins Problem 2 (A) Sketch and show this region in the X-Y Coordinate System. (B) Using "Disks"', calculate the "Volume of Revolution" by rotating the region about the X-Axis

Answer 1

(A) The area in figure shaded with parallel red straight lines shows the area bounded by y = 4x²+4, y = 0 (x-axis), x=10 and x=0 (y-axis) .

400 F 350 300 250 x=0 x=10 200 y-4x2 +4 150 100 y-4x+4 y=0 -20 -15 -10 0 10 15 20

(B) Disk Method states that the volume of the solid formed by revolving the region bounded by f(x) and $\small a \leqslant x \leqslant b$ about the x-axis is given by

$\small Volume = \pi \int_{a}^{b}(f(x))^2dx$

Using Disk method to calculate the volume of the region bound by y = 4x²+4, x=10 and x=0 about x-axis (i.e. y = 0),

$\small V = \pi \int_{x=0}^{10}(4x^2+4)^2dx$

    $\small = \pi \int_{x=0}^{10}(16x^4+32x^2+ 16)dx$

    $\small = \pi \left[ \frac{16x^5}{5}+\frac{32x^3}{3}+ 16x \right ]_0^{10}$

    $\small = \pi \left[ \frac{16}{5}(10^5-0)+\frac{32}{3}(10^3-0)+ 16(10-0) \right ]$

    $\small = 1039126.56$

    $\small \approx 1.0391\times 10^6$ [Ans]