An Introduction

to the

Fundamental Theorem of Calculus

In this worksheet we will try to find the value of

by using some limits of Riemann sums. In this worksheet, however, a and b will not have specific values.

You should be at the Calculus Menu of MenuMaple. Enter the expression assigned to you by your instructor and give it the name f. In this example we will use x^2 for f.

> f:=x^2;

Now select Riemann... and use the Left endpoint method (in the first column). Use f as the expression with x as the variable from a to b with n subdivisions. Use V as the name of the result.

> with(math191):

> V:=Lsum(f,x=a..b,n);

Note that V depends on a, b, and n. If a and b had actual values as they did in the worksheet on limits of Riemann sums, then V would look like the expressions obtained in that worksheet.

To find the value of the definite integral take the limit of V as n approaches infinity. Use the limit button with V as the expression, n as the variable, and approach infinity.

> limit(V,n=infinity);

After taking the limit we have an expression in a and b.

Now select Riemann... and use the Midpoint method (in the first column). Use f as the expression with x as the variable from a to b with n subdivisions. Use M as the name of the result.

> M:=Msum(f,x=a..b,n);

Now take the limit of M as n approaches infinity.

> limit(M,n=infinity);

Again we get an expression in a and b. In fact it is the same one we got using Left endpoint rectangles.

Now select Riemann... and use the Right endpoint method (in the first column). Use f as the expression with x as the variable from a to b with n subdivisions. Use R as the name of the result.

> R:=Rsum(f,x=a..b,n);

Now find the limit of R as n approaches infinity.

> limit(R,n=infinity);

Note that no matter which type of sum we used we got the same result. How is your answer related to your original function? Write down the result and the expression used for f. Return to the classroom and we will look at your results and the results of the other students in the class who were assigned different expressions for f.

Dan Symancyk November 1996 April 1998 June 1999

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