Mathematics: The region in the first quadrant bounded by the x-axis, the line x = ln(π), and the curve y = sin(e^x) is rotated about the x-axis. what is the volume of the generated solid?

The region in the first quadrant bounded by the x-axis,

Ответ:

lilpeepxliltracy

27.07.2019

The volume is approximately 2.498see diagram. if we graph these functions out with a graphing calculator, we see that this region also has to be bounded by the y-axis to keep it in the first quadrant.we can use discs to find the volume as there is no gap between the axis of revolution and the region.the radius of one disk is the distance between the curve y = sin(e^x) and the x-axis. this is just the y-coordinate given by the curve's equation, which is $\text{radius} = \sin(e^x)$ the area of one cross section uses the formula for the area of a circle: $\begin{aligned} a(x) & = \pi (\text{radius})^2 \\ & = \pi\left[ \sin(e^x)\right]^2 \\ & = \pi \sin^2( e^x ) \end{aligned}$ the region lies between x = 0 and x = ln(π) so we integrate with those as our limits to find the volume (we'll be using a graphing calculator to approximate this integral $\begin{aligned} v & =\int_0^{\ln(\pi)} a(x) dx \\ & =\int_0^{\ln(\pi)} \pi \sin^2( e^x ) \; dx \\ & \approx 2.497992 \\ & \approx 2.498 \end{aligned}$