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Problem 78 Hard Difficulty
Show that $ \displaystyle \int_0^\infty e^{-x^2}\ dx = \displaystyle \int_0^1 \sqrt{-\ln y}\ dy $ by interpreting the integrals as areas.
Answer
$\int_{0}^{\infty} e^{-x^{2}} d x=\int_{0}^{1} \sqrt{-\ln y} d y$
Topics
Integration Techniques
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Video Transcript
The problem is show that the integral from there to infinity each connective at squared the ax is they go to integral from their toe. One beautiful negative. How and why you? Why interpreting into girls as areas. Now look at the graph here. Area under the grow up under the bunch attitudes and necktie Black Square from there are two Infinity is this part. Is this real Andi the integral from there to infinity, into negative X squared the ax justice area of this part This computing as follows off a worry. Small Yeah, eggs. The hide is because Teo eat connective ex Claire So area can become computed as integral from their all to infinity eatin negative X squared DX. We have another way to compute this area. Just look, We're a small wine here, Andi. Height is Rico too. Things Why is he going to eat to Matthew X squared? So for worry small. Why? How and why is equal to negative X square. So axe is equal to negative callin wine until this one. There's a hide. It's just beautiful. Negative on wine. Now we can compute the area as a integral from zero to one on DH e y times the hide. This is really tough. Negative on why this is our source area off this region. So is too integral. Lt's our vehicle.
Topics
Integration Techniques
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