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Exercises computing volume by cross-sectional area.

The base of a solid region is bounded by the curves $x = 0$, $y = x^2$ and $y = x$. The cross sections perpendicular to the $x$-axis are squares. Compute the volume of the region.
• A typical square cross section has side length $L = \answer {x - x^2}$ and area $A = \answer {(x-x^2)^2}$.
• Possible numerical values of the $x$-coordinates of points in the base range from a minimum value of $x = \answer {0}$ up to a maximum of $x = \answer {1}$.
• To compute volume, integrate:
Find the volume of the region in three-dimensional space defined by the inequalities
• Cross sections perpendicular to the $z$-axis are squarerectangulartriangluar with length $\answer {1}$ in the $x$-direction and width $\answer {z^2}$ in the $y$-direction.
• The area of a $z$ cross section is $A(z) = \answer {z^2}$.
• To compute volume, integrate:
A right circular cylinder of radius $1$ and height $3$ is twisted along its axis so that the disk at height $z$ is centered on the axis $x = \cos (2 \pi z/3), y = \sin (2 \pi z/3))$, which corresponds to one full twist along the axis. Compute the volume of this twisted cylinder.
A certain three-dimensional region has a base in the $xy$-plane which is bounded above by the graph $y = 1-x^2$ and below by $y=0$. Slices perpendicular to the $y$-axis are equilateral triangles whose base lies in the $xy$-plane as well. Compute the volume of the region.

### Sample Exam Questions

(2018 Midterm 1) Compute the volume of the region in 3-dimensional space which satisfies the inequalities
$\displaystyle \frac {2}{3}$ $\displaystyle \frac {3}{4}$ $\displaystyle \frac {4}{5}$ $\displaystyle \frac {5}{6}$ $\displaystyle \frac {6}{7}$ none of these
(2019 Midterm 1) The inequality defines an ellipse in the $xy$-plane whose area is $\pi a b$ for any positive values of the constants $a$ and $b$. Compute the three dimensional volume of the region defined by (Hints won’t reveal until after you choose a response.)
$\displaystyle 4 \pi z^2$ $4 \pi$ $\displaystyle \frac {\pi }{4z}$ $\displaystyle \frac {\pi }{4}$ $\displaystyle {\pi z}$ $\displaystyle {\pi }$