- Introduction
- Definite and indefinite integrals with Maple
- Finding area
- Approximating area using Riemann Sums
- Accuracy
- Exercises

- Computing net or total distance traveled by a moving object.
- Computing average values, e.g. centroids and centers of mass, moments of inertia, and averages of probability distributions.

To compute the indefinite integral

with Maple:

> int(x^2,x);Note that Maple does

The command to use is:

> int(x^2,x=0..4);

> f := x-> -x^2+4*x+6; > g := x-> x/3+2; > plot({f(x),g(x)},x=-2..6); > a := fsolve(f(x)=g(x),x=-2..0); > b := fsolve(f(x)=g(x),x=4..6); > int(f(x)-g(x),x=a..b);

> rightsum(x^2,x=0..4,7); > evalf(rightsum(x^2,x=0..4,7)); > evalf(leftsum(x^2,x=0..4,10)); > evalf(middlesum(x^2,x=0..4,20));

All of the Maple commands described so far in this lab can include a third
argument to specify the number of subintervals. The default is 4
subintervals. The example below approximates the area under
from to using the `rightsum` command with 50,
100, 320 and 321 subintervals. As the number of subintervals
increases, the approximation gets closer and closer to the exact
answer. You can see this by assigning a label to the approximation,
assigning a label to the exact answer and taking their
difference. The closer you are to the actual answer, the smaller the
difference. The example below shows how we can use Maple to
approximate this area with an absolute error no greater than 0.1.

\begin{verbatim} > with(student): > exact := 4^3/3; > estimate := evalf(rightsum(x^2,x=0..4,321)); > evalf(abs(exact-estimate));

- For the function
over the interval , use the
`leftbox`command to plot the rectangular approximation of the area above the -axis and under with 6 rectangles. Estimate the area under the curve with a Riemann Sum using the formula for the left-endpoint rule and show that you get the same answer when using the`leftsum`command. - The exact area under
above the axis over the interval
can be found using a definite integral. Plot over the given interval. Use the approximations
`rightsum`and`middlesum`to determine the minimum number of subintervals required so that the estimate of this area has an error no greater than 0.001. Which method do you think is better? - Find the area of the region bounded by the curves and . Include a plot of the region bounded by the given curves.
- Find the area of the region bounded by the curves and . Include a plot of the region bounded by the given curves.

2017-09-07