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

Questions

1. 1.

   What is the volume of the solid generated by revolving the region bounded by $y = \sqrt{x}$, the line $x = 4$, and the $x$-axis about the $y$-axis?

   • A) $\frac{64\pi}{5}$
   • B) $\frac{96\pi}{5}$
   • C) $24\pi$
   • D) $\frac{128\pi}{5}$
2. 2.

   The base of a solid is the region bounded by the curve $y = 2\sqrt{\sin x}$ and the $x$-axis on the interval $[0, \pi]$. If cross-sections of the solid perpendicular to the $x$-axis are equilateral triangles, what is the volume of the solid?

   • A) 4
   • B) $\sqrt{3}$
   • C) $2\sqrt{3}$
   • D) $4\sqrt{3}$
3. 3.

   Water flows into an initially empty reservoir at a rate of $r(t) = 3t\sqrt{t^2 + 9}$ liters per minute, where $t$ is measured in minutes. How many liters of water accumulate in the reservoir during the first 4 minutes?

   • A) 49 liters
   • B) 84 liters
   • C) 98 liters
   • D) 125 liters
4. 4.

   What is the area of the region bounded by the curves $x = 3 - y^2$ and $x = y + 1$?

   • A) $\frac{7}{2}$
   • B) $\frac{9}{2}$
   • C) $\frac{11}{2}$
   • D) $\frac{16}{3}$
5. 5.

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

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

Answers and step-by-step solutions appear after you finish the test.