Applications of integrals (area, volume, accumulation) Knowledge-Based Download PDF

Questions

1. 1.

   What is the volume of the solid generated by revolving the region bounded by the curve $y = 3x - x^2$ and the $x$-axis about the $y$-axis?

   • A) $\frac{27\pi}{2}$
   • B) $\frac{27\pi}{4}$
   • C) $\frac{81\pi}{10}$
   • D) $27\pi$
2. 2.

   The base of a solid is the region enclosed by the curve $y = 2\sqrt{x}$, the horizontal line $y = 0$, and the vertical line $x = 3$. If every cross section of the solid perpendicular to the $x$-axis is an equilateral triangle with one side in the base, what is the volume of the solid?

   • A) $3\sqrt{3}$
   • B) $6\sqrt{3}$
   • C) $\frac{9\sqrt{3}}{2}$
   • D) $9\sqrt{3}$
3. 3.

   Water flows into a storage tank at a rate of $r(t) = 12\sqrt{t}$ gallons per hour, where $t$ is measured in hours. If the tank contains 50 gallons of water at $t = 0$, how many gallons of water are in the tank at $t = 4$ hours?

   • A) 64 gallons
   • B) 114 gallons
   • C) 74 gallons
   • D) 146 gallons
4. 4.

   What is the area of the region enclosed between the curves $y = x^2 - 2x$ and $y = x + 4$?

   • A) $\frac{115}{6}$
   • B) $\frac{121}{6}$
   • C) $\frac{125}{6}$
   • D) $\frac{131}{6}$
5. 5.

   What is the volume of the solid generated by revolving the region bounded by the curves $y = 2x$ and $y = x^2$ about the $x$-axis?

   • A) $\frac{32\pi}{15}$
   • B) $\frac{16\pi}{5}$
   • C) $\frac{64\pi}{15}$
   • D) $\frac{128\pi}{15}$