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Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral.

$ \displaystyle \int_0^1 \sqrt{x - x^2}\ dx $ ; entry 113

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

Missouri State University

Campbell University

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.

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Use the formula in the ind…

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Use the indicated entry in…

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Use the Table of Integrals…

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$1-4$ Use the indicated en…

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Okay, so this question wants us to evaluate this integral by using a formula from the table. So to do this, let's look at the formula on the table they want, which is this expression. So to do this, we need to turn our original expression into something that looks like this. So we see we have an X minus X squared, and we need to turn that into a to a U minus. You squared. So, to a has to equal one or a must be 1/2. So then plugging into the formula we see at the integral from 01 of X minus X squared is equal to Well, we just got a plug in 1/2 for a and we're evaluating this whole thing from 0 to 1 and then we'll see that this simplifies to 1/8 time's the co sign in verse of negative one, which is pie times on a or hi, divided by eight

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