Final answer:
To find the volume of the solid formed by revolving the region about the x-axis, we use the method of cylindrical shells. The definite integral that represents the volume becomes V=∫0c2πx(-x^4)dx.
Step-by-step explanation:
To evaluate the definite integral that represents the volume of the solid formed by revolving the region about the x-axis, we need to use the method of cylindrical shells. The equation y=-x^4 represents a parabola with its vertex at the origin. To find the volume, we will rotate the region bounded by the curve and the x-axis.
First, we need to determine the limits of integration. Since the curve is symmetric with respect to the y-axis, we only need to focus on the positive x-values. The curve intersects the x-axis at x=0, and we want to find the x-value where y=-x^4 reaches 0. So, set -x^4=0 and solve for x. We find that x=0. Therefore, the limits of integration are from x=0 to x=c, where c is the x-value where y=-x^4 reaches 0.
To find the volume of the solid, we use the formula V=∫2πxf(x)dx, where f(x) is the equation of the curve. In this case, f(x)=-x^4. So the definite integral that represents the volume is V=∫0c2πx(-x^4)dx.