Integrals 215 problem 07.mws

1 . Areas between curves.

In about the third century BC, Archimedes showed that the area of a parabolic segment bounded by a parabola and a line is 4/3 the area of the triangle ABC, where AB is the "base" of the parabolic segment and C is its vertex. Consider the parabolic segment bounded by the parabola y = x^2 and the line y = 1 .

> plot([x^2,1],x=-1..1,filled=true);

[Maple Plot]

Submission:

(a) Verify Archimedes' result in this case.

(b) Maple work that checks your work in (a).

(c) Archimedes didn't have Maple. Find out and answer the following questions.

i) Did Archimedes have a calculus book? Note that Archimedes did not have any paper.

Before about 100 BC Greeks and Romans wrote on papyrus or a codex.

ii) What are these things?

Submission worksheet:

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2. Checking your work with Maple, areas between curves.

Consider the region bounded by 4*x+y^2 = 0 , and y = 2*x+4 . We can find this area in two ways. One way is to integrate with respect to x, and another way is to integrate with respect to y.

Submission:

Submit the following

(a) A graph of the region.

(b) The Maple work leading to the area of the region obtained by integrating with respect to x, and

(c) the Maple work leading to the area of the region obtained by integrating with respect to y.

Submission worksheet:

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