Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis

Accepted Answer

First, identify the region R bounded by the curves: $$y = x$$^{2}$$, y = 2 - x$, and x = 0$ in the first quadrant. Sketching these curves helps visualize the area and the solid formed when revolving around the y-axis. Determine the points of intersection between y$$ = $$x^{2}$$ and $$y = 2 - x by $$setting $$x^{2} = 2 - x$. Solve this equation to find the x-values where the curves meet, which will serve as limits of integration. Since the solid is generated by revolving the region around the y-axis, decide on the method to use: the shell method is often convenient when revolving around the y-axis and the functions are given in terms of x. Set up the volume integral using the shell method formula: V$$ = \int_{a}^{b} 2\pi \cdot (	ext{radius}) \cdot (	ext{height}) \, dx$$. Here, the radius is the distance from the y-axis, which is x$, and the height is the vertical distance between the curves,  (2 - x$$) - $$x^{2}$$ $$. Write the integral explicitly as V$$ = \int_{x=a}^{x=b} 2\pi x \left( (2 - x) - $$x^{2}$$ ight) dx$$, where a$ and b$ are the intersection points found earlier. This integral represents the volume of the solid formed by revolving the region around the y-axis.

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis

Risposta video verificata per un problema simile:

Concetti chiave

Setting up the region bounded by curves

Methods of finding volumes of solids of revolution

Volume calculation about the y-axis using the shell method

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis

Risposta video verificata per un problema simile:

Concetti chiave

Setting up the region bounded by curves

Methods of finding volumes of solids of revolution

Volume calculation about the y-axis using the shell method

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis