   Chapter 7.8, Problem 78E

Chapter
Section
Textbook Problem

# Show that ∫ 0 ∞ e − x 2 d x = ∫ 0 1 − ln y   d y by interpreting the integrals as areas.

To determine

To show:

That 0ex2dx=01lnydy by interpreting the integrals as area.

Explanation

Consider the graph of ex2,

y=ex2

Take natural log both sides and simplify,

lny=x2x=lny

The graph of x=lny will be same as of y=ex2 in Ist quadrant, and won’t exist in IInd quadrant as  it’s domain is (0,).

Hence, area under both by both curves will be same in Ist quadrant.Area under x=lny in first quadrant is given by

01lnydy

And area under y=

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