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I hold a BS in computational mathematics from Rochester Institute of Technology. I worked as a computer programmer for five years before I became a homeschool parent. I homeschooled both my children from Kindergarten to graduation and both have gone on to the college of their choice! I have taught middle school and high school math in a homeschool co-op for ten years (pre-algebra through calculus AB). For the last two years I have also taught English as a second language to children in China through VIPKid.

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_1 $ about $ BC $

Find the volume of the described solid $ S $.The base of $ S $ is the same base as in Exercise 56, but cross-sections perpendicular to the x-axis are squares.

If the region shown in the figure is rotated about the y-axis to form a solid, use the Midpoint Rule with $ n = 5 $ to estimate the volume of the solid.

Let $ T $ be the triangular region with vertices $ (0, 0) $, $ (1, 0) $, and $ (1, 2) $, and let $ V $ be the volume of the solid generated when $ T $ is rotated about the line $ x = a $, where $ a > 1 $. Express $ a $ in terms of $ V $.

Use Definition 2 to find an expression for the area under the graph of $ f $ as a limit. Do not evaluate the limit.

$ f(x) = \sqrt{\sin x}, \hspace{5mm} 0 \le x \le \pi $

For the function $ f $ whose graph is shown, list the following quantities in increasing order, from smallest to largest, and explain your reasoning.

(A) $ \displaystyle \int^8_0 f(x) \, dx $(B) $ \displaystyle \int^3_0 f(x) \, dx $(C) $ \displaystyle \int^8_3 f(x) \, dx $(D) $ \displaystyle \int^8_4 f(x) \, dx $(E) $ f'(1) $