How to take a definite integral of an irrational-square function?

A

agentx0012014-02-12 20:35:13

Mathematics

agentx001, 2014-02-12 20:35:13

I got sick, missed most of the topic, and then bang (unexpectedly) today's test in algebra. Got a task: calculate the integral from -2 to 2 of the function sqrt(4-x^2) over dx. He got a little crazy, but after thinking a little, he himself realized that the problem can be solved geometrically: y = sqrt(4-x^2) -> y^2 = 4-x^2 -> x^2 + y^2 = 4 -- a after all, this is a function of a circle with a center at (0; 0) and a radius of 2. Well, then we remember that the integral is the area of the subgraph and elementarily solve. An ode to myself :)
But I'm still tormented by the question of how to take the antiderivative from sqrt (4-x ^ 2), in order to solve it with the Newton-Leibets formula?

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4 answer(s)

R

rafuck, 2014-02-12
@agentx001

the simplest is to make the substitution x = 2sin(t) (or x = 2cos(t))
then the reduction formula
and only then the Newton- Leibniz formula

A

Alexey Vanin, 2014-02-12
@jask

The antiderivative is calculated according to the tabular formula. Will be equal to arcsin(x/2). Further, given that the function is odd, we calculate. It should be Pi. But I can mess something up.

V

vans239, 2014-02-12
@vans239

winru.ru/kul4/kul22.html

E

Eugene, 2014-02-12
@Nc_Soft

Download Demidovich's problem book, there are not only tasks, but also methods for solving them.