Denition approach of learning new topics
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32 5 INTEGRATION<br />
d<br />
dx F (x) = f(x) then ´ f(x) dx = F (x)<br />
x=b ´<br />
Hence f(x) dx = F (x)| b a<br />
= F (b) − F (a)<br />
x=a<br />
Example. Evaluate<br />
2´<br />
x 2 dx<br />
We know ´ x 2 dx = x3<br />
3 + c 2<br />
1<br />
ˆ<br />
1<br />
x 2 dx = x3<br />
3<br />
= 23<br />
∣<br />
2<br />
1<br />
3 − 13<br />
3<br />
= 7/3<br />
Example. Find the area <strong>of</strong> the circle with radius R using integration<br />
Method I<br />
Area <strong>of</strong> the circle is integration <strong>of</strong> all small circles concentric with the given<br />
circle (as shown in the gure)<br />
We take a random such small thickness circle at a variable distance r from<br />
the center.<br />
6<br />
Now the area <strong>of</strong> the thin ring <strong>of</strong> dx thickness at a distance <strong>of</strong> r (random)<br />
from center = 2πr · dr<br />
Now we collect all such thin rings which are at a distance <strong>of</strong> zero(0) to R<br />
6 dx = lim ∆x so dx is an every decreasing and shrinking quantity. Hence the limit <strong>of</strong><br />
∆x→0<br />
the approximation gets us the integration to give the right answer.
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