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%Homework on Definite Integrals
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\begin{homework}\
\label{homework:17}

\medskip

\exercise %1.
In each case find the $n$-th left endpoint Riemann sum (for the indicated
value of $n$), and compare with the true value of the area obtained by the
anti-derivative method.  In each case draw a picture showing the area in
question and the rectangles in the Riemann sum, from which it should be
clear why the true value is greater or less than the Riemann sum.  

\smallskip

\begin{centering}
\begin{tabular}{lll}
(a) $\disp\int_0^2(4-x^2)dx$, $n=4$; &
(b) $\disp\int_0^2(4-x^2)dx$, $n=8$; &
(c) $\disp\int_0^4\sqrt{x}dx$, $n=4$; \\
(d) $\disp\int_0^{\pi/2}\cos x\,dx$, $n=3$;&
(e) $\disp\int_0^{\pi/2}\cos x\,dx$, $n=6$. &
\end{tabular}\\
\end{centering}


\smallskip

\exercise %2.
Compute: (a) $\sum_{i=1}^5 i^2$; (b) $\sum_{i=0}^3\frac{1}{ 1+i}$; (c)
$\sum_{i=0}^3\cos(i\frac{\pi}{ 6})$.

\smallskip

\exercise %3.
Find the area under 

\smallskip

\begin{centering}
\begin{tabular}{ll}
(a) $x^2+x$ between 1 and 4; & 
(b) $\sqrt{4-3x}$ between 0 and 1;\\
(c) $1/\sqrt{4-3x}$ between 0 and 1;&
(d) $1/(2x+1)^3$ between 0 and 2; \\
(e) $\cos(\pi x)$ between $-1/2$ and $1/2$.&
\end{tabular}\\
\end{centering}

\smallskip

\exercise %4.
Find the area contained 

(a) between the curve $y=\sqrt{1-x}$ and the coordinate axes;

(b) under both $\sin x$ and $\cos x$ in the region from $x=0$ to $x=\pi/2$;

(c) between the curve $y=\cos(\frac{\pi}{ 2}x)$ and the curve $y=x^2-1$;

(d) between the line $y=2$ and the branch of the curve $y=\sec^2x$ lying
between the asymptotes at $\pm\pi/2$.

\smallskip

\exercise %5.
Evaluate the following definite integrals: 

\begin{centering}
\begin{tabular}{lll}
(a) $\disp\int_1^3\sqrt{4x-3}dx$; &
(b) $\disp\int_{-1}^1(x^2+1)^2dx$;&
(c) $\disp\int_{-1}^1(x^3+x)^3dx$;\\
(d) $\disp\int_0^{0.5}\sin^3(\pi x)\cos(\pi x)dx$; &
(e) $\disp\int_0^{\pi/4}\frac{\sin x\,dx}{ \cos^5x}$;& 
(f) $\disp\int_0^3 x\sqrt{9-x^2}dx$.
\end{tabular}\\
\end{centering}

\smallskip

\exercise %6.
Find the normal number of degree-days  in Fairbanks (a)
during an entire year, (b) during the coldest 2-month period (i.e., the
time period extending 30 days before and 30 days after the coldest day),
and (c) during the warmest 2-month period.  Assume that the inside
temperature is maintained at $65^\circ$F, and use the formula for the
normal mean daily temperature that is given in Problem 6 of the homework
set on trig functions.

\smallskip

\exercise %7.
In each case, find formulas and draw graphs for $a(t)$ and $v(t)$, and
find the net distance traveled during the given time interval.
(a) The initial velocity is zero.  The acceleration starts at $0$,
increases linearly to 4 units/sec${}^2$ during the first two seconds,
remains constant at 4 units/sec${}^2$ for $2\le t\le 4$, and drops off
linearly to $0$ during the last 4-second period $4\le t\le 8$.  (b) The
initial velocity is zero.  The accleration is the minimum of $\sin t$
and $1-t$ for $0\le t\le 1$.  (Use Newton's method to find the point of
intersection of these two functions, maintaining four significant figures
of accuracy throughout the problem.)

\end{homework}
