Let be the region in the -plane bounded by the graphs of • • and the vertical lines and

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

Problem

Let

R

be the region in the

xy

-plane bounded by the graphs of •

y = -x^2 + 4

•

y = x

and the vertical lines

x = \dfrac{-1 - \sqrt{17}}{2}

and

x = \dfrac{-1 + \sqrt{17}}{2}

. The region

R

is revolved about the horizontal line

y = -3

, producing a solid. (1) Part 1: (a) Verify algebraically that the curves

y = -x^2 + 4

and

y = x

intersect at

x = \dfrac{-1 \pm \sqrt{17}}{2}

. (2) Part 2: (b) Express the volume

V

of the solid as the sum of one or more definite integrals with respect to

x

. Clearly indicate the integrand on each subinterval and the corresponding limits of integration. (3) Part 3: (c) Evaluate the integral(s) found in part (b) to obtain an exact value for

V

Related theory

Area Between Curves
Area Applications
Volume Applications
Applications of Integration

Browse the problem library