Page 251 - Calculus Workbook For Dummies

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Chapter 12: Who Needs Freud? Using the Integral to Solve Your Problems

1
3
4. Find the area on the right — t is on top — then add that to .
12
1 1
4
6
5
3
Area= # _ t - t i = 1 t - 1 t E = 1 - 1 = 1
4 6 4 6 12
0 0
1 1 1
Therefore, the total area is + , or .
12 12 6
5
3
Note that had you observed that both t and t are odd functions, you could have reasoned
that the two areas are the same, and then calculated just one of them and doubled the result.
*e The lines y = x, y = 2x – 5, and y = –2x + 3 form a triangle in the first and second quadrants.
What’s the area of this triangle? The area is 6.
1. Graph the three lines.
2. Find the three points of intersection.
a. y = x intersects b. y = x intersects c. y = 2x – 5 intersects
y = 2x – 5 at x = 2x – 5 y = –2x + 3 at x = –2x + 3 y = –2x + 3 at 2x – 5 = –2x + 3
x = 5 and, thus, y = 5 x = 1 and, thus, y = 1 x = 2 and, thus, y = –1

3. Integrate to find the area from x = 1 to x = 2; y = x is on the top and y = –2x + 3 is on the
bottom, so
2
^
Area= # ` x - - 2 x + 3hj dx
1
2
= 3 #^ x - 1h dx
1
2
1 2
x
= 3 ; x - E
2
1
1 3
1 =
3 2 - h
= = ^ 2 - c - mG
2 2
4. Integrate to find the area from x = 2 to x = 5; y = x is on the top again, but y = 2x – 5 is on the
bottom, thus
5
Area= # ` x - ^ 2 x - 5hj dx
2
5
= #^ - x + 5h dx
2
5
1
2
= - x + 5 xE
2
2
25 9
= - + 25 - - + 10 =
2
h
^
2 2
Grand total from Steps 3 and 4 equals 6.
Granted, using calculus for this problem is loads of fun, but it’s totally unnecessary. If you cut
the triangle into two triangles — corresponding to Steps 3 and 4 above — you can get the total
area with simple coordinate geometry.