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\large 
\underline
{MA2160 \hspace{5mm}Calculus II with Technology \hspace{50mm} Ch. 8-1}

We can use the definite integral to find areas and volumes by applying a Riemann sum.

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Goal
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\begin{enumerate}
\item Apply a Riemann sum to estimate an area
\item Apply a definite integral to the Riemann sum to find exact area

\end{enumerate}
Recall that we estimated area under the curve by drawing rectangles that approximated the area under the curve, and that we increased the accuracy of our estimate by taking the limit of the width of interval as it approaches 0.  We can use the same approach to find the areas and volumes of a wide variety of shapes and figures. Admittedly, we know the formulas for many shapes from geometry, but our shape will not always be so nice 

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\begin{Large}
Steps to find Area
\end{Large}
\begin{enumerate}
\item Determine orientation of the slice (or rectangle)--horizontal or vertical
\item If possible and helpful, write an equation that matches the shape analyzed and/or place on coordinate axes
\item Find area (or volume) of one slice.
\item Convert formula to variable that matches the slice
\item Write definite integral
\item Find definite integral
\end{enumerate}
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\small

\begin{enumerate}
\item A right triangle has a height of 10 and a hypotenuse of length 20.  Find the area taking horizontal and then vertical slices.
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\item Find the area between the functions $9-x^2$ and $(x-3)^3$ from $x=0$ to $3$.
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\item Find the area of a circle with radius 5 using vertical slices.
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\item Given that the height of a cone is three times the radius, find the volume of a cone 15 $cm$ tall taking horizontal slices.
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\item Given that the side of the base of a square pyramid is the same as the height, find the volume of a square pyramid whose base is 100 $m$ on a side.
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\item The cross-section of a 100$m$ tunnel is a semi-circle with radius 50$m$.  Find the volume of the tunnel using vertical and horizontal slices.
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\end{enumerate}
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