Numerical Integration

Many integrals (such as $\int_{{0}}^{{1}}$e-x2dx and $\int_{{0}}^{{\pi }}$${\frac{{%
\sin t}}{{t}}}$dt) cannot be evaluated exactly, but you get numerical approximations by choosing Evaluate Numerically.

$\blacktriangleright$ Evaluate Numerically

$\dint_{{0}}^{{1}}$e-x2dx = .7468241328 6pt

$\dint_{{0}}^{{\pi }}$${\dfrac{{\sin t}}{{t}}}$dt = 1.851937052

Integrals associated with arc lengths of curves can almost never be evaluated exactly.

Given a curve y = f (x), the arc length between x = a and x = b is given by the integral

$\displaystyle \int_{{a}}^{{b}}$$\displaystyle \sqrt{{1+\left( f^{\prime }(x)\right) ^{2}}}$dx


For example, given f (x) = x sin x, consequently, f(x) = sin x + x cos x, you can find the length of the arc between x = 0 and x = π by applying Evaluate Numerically.

$\blacktriangleright$ Evaluate Numerically

$\dint_{{0}}^{{\pi }}$$\sqrt{{1+\left( f^{\prime }(x)\right) ^{2}}}$dx = $\dint_{{0}}^{{\pi }}$$\sqrt{{1+\left( \sin x+x\cos x\right) ^{2}}}$dx = 5.04040692

Curves in the plane or three-dimensional space can also be represented parametrically.



\begin{example}
The center of a piece of string of radius $r$\ wound around a s...
...y
well if the string is wrapped around the spool numerous times.
\end{example}

The preceding example can be visualized graphically. In the Plot Properties tabbed dialog, set the Plot Style to Hidden Line, the Sample Size to 99, the Radius to 1/8, and the Domain Interval to 0≤t≤16π = 50.265; choose Equal Scaling on Each Axis.

$\blacktriangleright$ Plot 3D + Tube

$\left[\vphantom{ \frac{9}{8}\cos t,\frac{9}{8}\sin t,\frac{t}{8\pi }}\right.$${\frac{{9}}{{8}}}$cos t,${\frac{{9}}{{8}}}$sin t,${\frac{{t}}{{8\pi }}}$$\left.\vphantom{ \frac{9}{8}\cos t,\frac{9}{8}\sin t,\frac{t}{8\pi }}\right]$

dtbpF3in2.0003in0pt