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 axis.y=√sin x,y=1, and x=0; about the x-axis

Accepted Answer

First, identify the region R bounded by the curves: $$ y = \sqrt{\sin x} $$, $$ y = 1 $$, and $$ x = 0 . $$Determine the interval for $$ x $$ where these curves intersect and define the region. Since $$ y = \sqrt{\sin x}  is $$defined where $$ \sin x \geq 0 $$, consider $$ x  in$$ $$ [0, \pi] $$ because $$ \sin x  is $$positive there, and find the point where $$ \sqrt{\sin x} = 1  to $$find the upper bound of $$ x . $$Next, sketch or visualize the region and the axis of rotation, which is the x-axis. Since the region is bounded above by $$ y = 1 $$ and below by $$ y = \sqrt{\sin x} $$, revolving around the x-axis will create a solid with a hole (washer) at each cross-section. Set up the volume integral using the washer method. The volume $$ V  is $$given by integrating the area of the washers perpendicular to the x-axis over the interval found in step 1. The outer radius $$ R_{outer}  is $$the distance from the x-axis to $$ y = 1 $$, and the inner radius $$ R_{inner}  is $$the distance from the x-axis to $$ y = \sqrt{\sin x} . So$$, the volume integral is:

$$ V = \pi \int_{0}^{a} \left( R_{outer}^2 - R_{inner}^2 \right) \, dx = \pi \int_{0}^{a} \left(1^2 - (\sqrt{\sin x})^2 \right) \, dx $$

where $$ a  is $$the upper limit of integration found in step 1. Simplify the integrand inside the integral. Since $$ (\sqrt{\sin x})^2 = \sin x $$, the integrand becomes $$ 1 - \sin x . So $$the volume integral simplifies to:

$$ V = \pi \int_{0}^{a} (1 - \sin x) \, dx $$ Finally, evaluate the integral (without calculating the numerical value here). Use the antiderivative of $$ 1 - \sin x $$, which is $$ x + \cos x $$, and apply the limits from 0 to $$ a . $$This will give the expression for the volume of the solid generated by revolving the region about the x-axis.

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

y=√sin x,y=1, and x=0; about the x-axis

Risposta video verificata per un problema simile:

Concetti chiave

Volume of Solids of Revolution

Washer Method

Setting up Integration Limits and Radii