The region is bounded above by the graph of and below by the –axis on the closed interval

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

The region

R

is bounded above by the graph of

y = 2x - x^2

and below by the

x

–axis on the closed interval

[0,2]

. When

R

is revolved about the horizontal line

y = -1

, a solid

S

is generated. (a) Determine the

x

–coordinates of the points where the curve

y = 2x - x^2

intersects the

x

–axis. (b) Write, but do not evaluate, a definite integral that represents the volume of the solid

S

. (c) Evaluate the integral from part (b) to obtain an exact expression for the volume of

S

. (d) Approximate the volume found in part (c) to three decimal places.