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Find the volume of the solid obtained when the region under the curve $ y = \arcsin x $, $ x \ge 0 $, is rotated about the y-axis.

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

Campbell University

Baylor University

University of Michigan - Ann Arbor

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.

07:39

Find the volume of the sol…

04:49

02:53

The region under the curve…

01:50

$\begin{array}{l}{\text { …

02:52

04:17

02:09

03:44

01:05

02:30

Okay, This question Once a stir of all of the area underneath Weikel sine inverse X, where X is greater than zero around the y axis. So, with all volume of revolution questions, let's draw to see what we're dealing with. So Arc Sine has a domain from negative one No one. So if we're considering the positive side, we end at one here. And then if we graph park sign of X, it looks something like this, but we're evolving this area around the Y axis. So what we can do is convert this to a function of X and consider this area right here. And remember this Y value is pirate, too, because arc sine of one is pirates. So to convert to a function of acts, we could just see that sign of why equals X. So that becomes are integral. So are equals. Sign Why? So that means air volume is pi times the integral from zero to pi over too of sign of why squared d y and no, that's an integral that we can look up. Or at this point, you may have it memorized. So the integral of sine squared is 1/2 why minus 1/4 sign of two. Why? From 0 to 2. Pi Sorry, zero pi over too. And this becomes pi Times pi over two divided by two minus Well signed pie is zero minus 1/2 time. Zero minus 1/4 time. Zero. So this is just equal to pi times pi over two divided by two, which is equal to high times over for or pi squared over for so that's our final answer.

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