Shell Method Examples

Find the area bounded by the following curves/lines

$y = x + 1$

$y = 9 - x^2$

$x = -1$

$x = 2$

Find the area between the curves $y = x$ and $y = x^2$

Compute the area between $y = \sin{x}$ and $y = \cos{x}$ and the interval $[\frac{\pi}{4}, \frac{5\pi}{4}]$

Let R be the region enclosed by the graph of $f(x) = x^4 - 2.3x^3 + 4$ and the horizontal line y = 4, as shown in the figure above.

A. Find the volume of the solid generated when R is rotated about the horizontal line y = -2

B. Region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in R. Find the volume of the solid.

C. The vertical line x = k divides R into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value k.