.

Back to the class
Section 7.1 #81: Find the area between the graph of $y=\sin(x)$ and the line segment joining the points $(0,0)$ and $\left( \dfrac{7\pi}{6}, - \dfrac{1}{2} \right)$.
Solution: Draw this:

The line segment has slope $\mathrm{slope}=\dfrac{-\frac{1}{2}-0}{\frac{7\pi}{6}-0} = -\dfrac{3}{7\pi}.$ So the line segment obeys the equation $y-0=-\dfrac{3}{7\pi}(x-0),$ or more simply, $y=-\dfrac{3}{7\pi}x$. Now compute
$$\begin{array}{ll} \displaystyle\int_0^{\frac{7\pi}{6}} \sin(x) - \left( - \dfrac{3}{7\pi} \right)x \mathrm{d}x &= -\cos(x) + \dfrac{3}{14\pi}x^2 \Bigg|_0^{\frac{7\pi}{6}} \\ &= \left( -\cos \left( \dfrac{7\pi}{6} \right) \right)+ \dfrac{3}{14\pi} \left( \dfrac{7\pi}{6} \right)^2 - \left( -\cos(0) + \dfrac{3}{14\pi}0^2 \right) \\ &=- \left( -\dfrac{\sqrt{3}}{2} \right)+\dfrac{147\pi^2}{504\pi} - (-1) \\ &=1 + \dfrac{\sqrt{3}}{2} + \dfrac{7\pi}{24}. \end{array}$$

Section 10.1 #64: Sketch how a plane can intersect the cone to get the following conic sections:
a.) circle,
b.) ellipse,
c.) parabola, and
d.) hyperbola.
Solution: