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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int_0^2 x^2 \sqrt{4 - x^2}\ dx $

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$\pi$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

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University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

01:53

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

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

02:06

Use the Table of Integrals…

05:43

03:15

07:55

03:06

03:50

Use the Table of Integral…

04:20

02:48

02:16

Okay, so this question wants us to answer this integral using a tape. So if we look at our table, we find this formula, which is very close to what we're given. So we just want to transform our integral into something that looks like this. So, as you can see, there is nothing to do to this. It's already exactly in this form, we just see that you equals X and A equals two. So this means that d U equals DX. So you don't have to worry about changing our limits or any general factors. And we also know our values. A So that means we can just integrate. So are integral is equal to plugging into our formula. Here you, over eight times to you squared minus a squared, which is four times the square root of you squared minus four plus ace eight of the fourth over eight, which is 16 over eight. Sign in verse of you over, too. And we're going from zero two. So if we plugger limits in here, we get to over a times a minus four times square root of four minus four plus to sign in verse of to over two. So that's our plus limit. And then are negative. Limit turns into zero plus to sign in verse of zero, which is also zero. So that means our answer just becomes to sign in verse of one, which is two times pi over two or pie.

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