Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.y=x and y=1+x/2; about y=3

Accepted Answer

First, identify the region R bounded by the curves $$ y = x $$ and $$ y = 1 + \frac{x}{2} . To do $$this, find the points of intersection by setting the two equations equal: $$ x = 1 + \frac{x}{2} . $$Solve the equation $$ x = 1 + \frac{x}{2} $$ for $$ x  to $$find the limits of integration. This will give the $$ x -$$values where the two curves intersect, which define the interval over which the region R lies. Since the solid is generated by revolving the region about the line $$ y = 3 $$, use the washer method. The radius of each washer is the distance from the line $$ y = 3  to $$the curve. Express the outer radius $$ R(x) $$ and inner radius $$ r(x)  as $$functions of $$ x $$:

$$ R(x) = 3 - y_{	ext{lower}}(x), \quad r(x) = 3 - y_{	ext{upper}}(x) $$

where $$ y_{	ext{lower}}(x) $$ and $$ y_{	ext{upper}}(x) $$ are the lower and upper curves respectively on the interval. Set up the volume integral using the washer method formula:

$$ V = \pi \int_{a}^{b} \left[ R(x)^2 - r(x)^2 ight] \, dx $$

where $$ a $$ and $$ b $$ are the intersection points found in step 2. Evaluate the integral to find the volume of the solid. This involves squaring the radii expressions, subtracting, and integrating with respect to $$ x $$ over the interval $$ [a, b] .$$

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.

y=x and y=1+x/2; about y=3

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Finding the Region Bounded by Curves

Volume of Solids of Revolution Using the Washer Method

Adjusting Radii for Revolution About a Line y = k