[Technology Exercise] 75. Find, to two decimal places, the x-c...

Question

[Technology Exercise] 75. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = arctan(x), and the line x = √3.

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Hello. In this video we are told that a region in the first quadrant is enclosed by the x axis. The curve y is equal to tangent inverse of 2 x, and the vertical line x is equal to 1. We want to determine the x coordinate of its centroid accurate to two decimal places by using the given equation where the integral from 0 to 1 of x tangent inverse of 2 x d x is equal to 5/8 tangent inverse of 2 minus 1/4. So, in order to approach this problem, we want to go ahead and use the given equation. The equation for the x coordinate of the centroid is given to us as a ratio of definite integrals. The numerator is the integral from 0 to 1 of X, tangent inverse of 2 X D X. And the denominator is the integral from 0 to 1 of tangent inverse of 2 x dx. Now here, the problem has worked out the value of the numerator for us. So what we need to do is we need to find the value of the denominator. So the first thing we want to do is we want to go ahead and evaluate the integral in the denominator. So again, that integral is going to be the integral from 0 to 1. Of tangent inverse of 2 x d x. Here what we can go ahead and do is we can approach this problem by using integration by parts. Recall that integration by parts is an integral of the form UDV and that can be rewritten as U multiplied by V minus the integral of V multiplied by DU. Here we are going to set u to equal the tangent inverse of 2 x. And DV is going to equal to DX. If we were to take the derivative of U, DU is going to equal to 2 divided by 1 + 4 x 2. And by taking the antiderivative of the DV statement, we get V is equal to x. So now if we use the identity for the integration by parts, we get the following integral. We get u multiplied by V, which is going to be x tangent inverse. Of 2 x which will be evaluated from 0 to 1 minus the definite integral from 0 to 1 of v multiplied by duu, which is 2 x divided by 1 + 4 x squared x. Now here, what we can go ahead and do is we can go ahead and apply a substitution to simplify the integral. Now, if we were to apply a substitution, let's say U is equal to 1 + 4 X2, then DU is equal to 8 XDX, which will give us 1/8 DU is equal to XDX. So if we were to apply the substitution and rewrite the given integral, we will have x equal or x multiplied by tangent inverse of 2 x evaluated from 0 to 1. -2/8, which simplifies to 1/4 of a definite integral from 8 to b of 1 divided by u d u. And this will further simplify to be x multiplied by the tangent inverse of 2 x evaluated from 0 to 1 minus 1/4 multiplied by the natural logarithm of U evaluated from a boundary from A to B. Now, if we were to resubstitute our substitution of U, this will further simplify to be x multiplied by the tangent inverse of 2 x evaluated from 0 to 1. -1/4 multiplied by the natural logarithm of 1 + 4 x 2 evaluated from 0 to 1. Now since the entire thing is being evaluated from 0 to 1, we can just go ahead and treat this as one giant antiderivative, but if we were to evaluate from 0 to 1, this entire thing will simplify to be tangent inverse of 2. 1/4 multiplied by the natural logarithm of 5. So this is going to be the value of the denominator. And now that we have the value of the denominator, by taking the value of the numerator and plugging these values back into the formula for the x coordinate of the centroid, we will get the following equation. We will get x bar is equal to 5/8 multiplied by the tangent inverse of 2 minus 1/4 divided by tangent inverse of 2 minus 1/4 multiplied by the natural logarithm of 5. Now at this point, we're going to plug this into a calculator, and this is going to approximate to give us an x coordinate of 0.63. This is going to be the solution to the problem, with the overall solution being option C. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

[Technology Exercise] 75. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = arctan(x), and the line x = √3.

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