Question
A region in the first quadrant is enclosed by the graphs of \(y=e^{2x}\),x = 1, and the coordinate axes.If the region is rotated about the y-axis, the volume of the solid that is generated is represented by which of the following integrals?
(A) \(2\pi\int_{0}^{1}xe^{2x}dx\)
(B)\(2\pi\int_{0}^{1}e^{2x}dx\)
(C)\(\pi\int_{0}^{1}e^{4x}dx\)
(D)\(\pi\int_{0}^{e}y ln ydy\)
(E)\(\frac{\pi }{4}\int_{0}^{e}ln^{2}ydy\)
▶️Answer/Explanation
Ans:A
Question
The volume of the solid obtained by revolving the region enclosed by the ellipse\( x^{2}+9y^{2}=9\) about the x-axis is
(A) 2π (B) 4π (C) 6π (D) 9π (E) 12π
▶️Answer/Explanation
Ans:B
Question
The region enclosed by the x-axis, the line x = 3, and the curve \(y=\sqrt{x}\) is rotated about the x-axis. What is the volume of the solid generated?
(A)\(3\pi\) (B)\(2\sqrt{3\pi} \) (C)\(\frac{9}{2}\pi\) (D)\(9\pi\) (E)\(\frac{36\sqrt{3}}{5}\pi\)
▶️Answer/Explanation
Ans:C
Question
If the region enclosed by the y-axis, the line y =2 , and the curve \(y=\sqrt{x}\) is revolved about the y-axis, the volume of the solid generated is
(A) \(\frac{32π}{5}\) (B) \(\frac{16π}{3}\) (C) \(\frac{16π}{5}\) (D) \(\frac{8π}{3}\) (E) π
▶️Answer/Explanation
Ans:A
