Chapter 8 and 9 Take-Home Test

Hello everyone,

Here is your take-home test covering the material from Chapters 8 and 9. This test assesses your understanding of key concepts we've covered, including applications of integration such as finding surface areas, hydrostatic forces, centroids, and differential equations.

Due Date: Tuesday, March 24th by 5:00 PM

Submission Instructions:

  • Please submit your completed test via email to Tony.Baker@ct.gov.
  • You can submit your answers as clear and readable pictures or as a scanned PDF document. Please ensure your submission is easy to read for accurate grading.

Important Notes:

  • There will be no class on Thursday, giving you ample time to work on the test. Use this time wisely!
  • Remember to show all your work. Partial credit will be awarded based on demonstrated understanding.

Test Questions:

  1. Surface Area of Revolution: Find the exact area of the surface obtained by rotating the curve $y = \frac{x^3}{6} + \frac{1}{2x}$ about the x-axis for $\frac{1}{2} \leq x \leq 1$. Remember the formula for surface area of revolution: $S = 2\pi \int_a^b y \sqrt{1 + (\frac{dy}{dx})^2} dx$.
  2. Hydrostatic Force: A swimming pool is 20ft wide and 40 feet long, and its bottom is an inclined plane. The shallow end has a depth of 3 ft, and the deep end is 12 ft. If the pool is full of water, find the hydrostatic force on:
    • (a) The shallow end side
    • (b) The deep end side
    • (c) One of the sides of the pool
    • (d) The bottom of the pool. Recall that Hydrostatic Force is given by $F = \int_a^b \rho h(y) w(y) dy$ where $\rho$ is the density of the fluid, $h(y)$ is the depth as a function of y, and $w(y)$ is the width as a function of y.
  3. Centroid: Find the centroid of the region bounded by the curves $x + 2y = 4$ and $x = y^2$. Remember that the centroid $(\bar{x}, \bar{y})$ is calculated by $$\bar{x} = \frac{1}{A} \int_a^b x(f(x) - g(x)) dx$$ $$\bar{y} = \frac{1}{A} \int_a^b \frac{1}{2} (f(x)^2 - g(x)^2) dx$$ where A is the area of the region.
  4. Mixing Problems: A vat with 500 gallons of beer contains 3% alcohol (by volume). Beer with 7% alcohol is pumped into the vat at a rate of five gal/min, and the mixture is pumped out at the same rate.
    • What is the percentage of alcohol after an hour?
    • How long will it take to have a beer with 6.5% alcohol?
    Remember the differential equation is of the form $dy/dt = rate_{in} - rate_{out}$.
  5. Orthogonal Trajectories: Find the orthogonal trajectories of the family of curves $y = \frac{1}{2x + k}$. Recall that to find orthogonal trajectories, first find $\frac{dy}{dx}$, then take the negative reciprocal. Solve the resulting differential equation.

Good luck with the test! Remember to review your notes and examples from class. Feel free to reach out if you have any clarifying questions before attempting the test.

Best regards,

Professor Baker