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Problem

Find the value of the constant $ C $ for which th…

14:02

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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.

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Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

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01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

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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.

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