Question

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

y = -2

Accepted Answer

First, identify the region R bounded by the curves: $$y = x$$^{2}$$, x = 1$, and y = 0$. This region lies between x=0$ and x=1$ since y=0$ is the x-axis and y$$=$$x^{2} is a $$parabola opening upwards. Since we are revolving the region about the line $$y = -2$$, which is horizontal and below the x-axis, we will use the shell method with vertical shells. The shells will be formed by vertical slices of the region at position $$x$$ with thickness $$dx. $$The height of each shell is the vertical distance between the curves $$y = x$$^{2}$$ (top) and y = 0$ (bottom), so the height is h(x$$) = $$x^{2} - 0$$ = $$x^{2}. $$The radius of each shell is the distance from the slice at $$y to $$the axis of rotation $$y = -2. $$Since the shell is at height $$y$$ between $$0$$ and $$x^{2}$$, the radius is the vertical distance from $$y to$$ $$-2. $$However, because we are integrating with respect to $$x$$, the radius is the distance from the shell at $$y = x$$^{2}$$ to y = -2$, which is r(x$$) = $$x^{2} - (-2$$) = $$x^{2} + 2$. The volume of the solid is given by the integral using the shell method formula: V = 2$$\pi \int_{a}^{b} (	ext{radius})(	ext{height}) \, dx$$. Substitute the expressions for radius and height and the limits a=0$ and b=1$:

V = 2$$\pi \int_{0}^{1} $$(x^{2}$$ + $$2)(x^{2}$$) \, dx$$.

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

y = -2

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39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.y = -2

Risposta video verificata per un problema simile:

Concetti chiave

Shell Method for Volume

Setting up the Shell Radius and Height

Region Boundaries and Limits of Integration

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

y = -2