5559 These exercises reference the Theorem of Pappus:If R is a bounded plane region and

Chapter 14, Problem 55

(choose chapter or problem)

55–59 These exercises reference the Theorem of Pappus: If \(R\) is a bounded plane region and \(L \)is a line that lies in the plane of \(R\) such that \(R\) is entirely on one side of \(L\), then the volume of the solid formed by revolving \(R\) about \(L\) is given by

volume = (area of \(R\)) · (distance traveled by the centroid)

Perform the following steps to prove the Theorem of Pappus:

(a) Introduce an xy-coordinate system so that \(L\) is along the y-axis and the region \(R\) is in the first quadrant. Partition \(R\) into rectangular subregions in the usual way and let \(Rk\) be a typical subregion of \(R\) with center \(\left(x_{k}^{*}, y_{k}^{*}\right) \text { and area } \Delta A_{k}=\Delta x_{k} \Delta y_{k}\). Show that the volume generated

by \(Rk\) as it revolves about \(L\) is

\(2 \pi \times x_{k}^{*} \Delta x_{k} \Delta y_{k}=2 \pi x_{k}^{*} \Delta A_{k}\)

(b) Show that the volume generated by \(R\) as it revolves

about \(L\) is

\(V=\iint_{R} 2 \pi x d A=2 \pi \cdot \bar{x} \cdot[\text { area of } R]\)

Equation  Transcription:

text transcription:

(x^8 _k, y^* _k)

Triangle A_k=triangle x_k, triangle y_k

2 pi x x^*_k triangle x_k triangle y_k=2 pix x_k^* triangle A_k

v= double integral_R 2pix dA= 2pi.x bar. [area of R]