Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 4 − 4x2, y = 0

volum-of-the-solid
y-4−4x2

#1

Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 4 − 4x2, y = 0

Answer:

We have to sum up the area of each infinitesimally small circular cross-sectional area in the solid in order to find volume. The radius of this circle at any point is given by the formula r = y = 4 - 4x^2. The formula for area, then, is A = pi * r^2, or pi * (r - rx^2)^2

To sum up all areas, integrate from -1 to 1, the bounds of the area enclosed by the graph (the zeros of the equation for y).

So we take the integral of pi * (4 - 4x^2)^2 from -1 to 1, and we get about 53.617