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
and the line
.

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

**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 , and . 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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