Graphing an Integral

Calculus II Project

Objective:

To find and graph an integral of a function whose antiderivative is not found by formulas in a Calculus II course.

To use the graphing capabilities of Maple.

Solved Problem:

Find the antiderivative, g( x ) , of f( x ) = sin on [0, 2 ].

(a) Verify that g'( x ) = f( x )

(b) Plot the graph of g( x ), the antiderivative of f( x ).

(c) Evaluate g(.1) approximately.

Solution:

The integral of f( x ) = sin cannot be found by any formulas. But by the Fundamental Theorem of Calculus and knowing that the graph of sin over any finite interval lies between -1 and 1, we see that the integral exists and is given by

We define g( x ) = integral sin( t^2)dt from 0 to x .

(a) Verify that g'(x) = f(x).

> with(plots):

> f := x -> sin(x^2);

> g := x -> int(sin(t^2), t = 0..x);

> diff(g(x),x);

(b) Graph of g(x).

> plot(g(x), x = 0..2*Pi);

(c) Evaluation of g(.1)

> g(.1);

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ASSIGNMENT

Problem 1:

(a) Find integral of sin (sqrt( x )) and check your answer. (This integral can be found by using int ).

(b) Plot the graph of integral of sin(sqrt( x )).

Problem 2:

(a) Find g( x ) = integral of exp(-.5* x ^2) and check your answer.

(b) Plot the graph of integral of exp(-.5*x^2)

(c) Evaluate g(2).

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MSIP Grant # P120A80089-98: "Three Urban Calculus Reform Programs: Adopting the Best" 1998-2001