41–48. Geometry problems Use a table of integrals to solve the...

Question

41–48. Geometry problems Use a table of integrals to solve the following problems.

46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says to calculate the area under the curve, Y is equal to 1 divided by the square root of quantity of X2 + 6 X + 10 in quantity from X equal to -3 to X equal to 2. Now we're asked to find the area underneath the curve Y, and the area underneath the curve Y is given by the integral of Y with respect to X. So we're going to need to evaluate the integral from minus 3 to 2 of Y, which is 1 divided by the square roots of the quantity of X2 + 6 X plus 10 in quantity DX. And so, in order to get this into a form that we can actually evaluate, what we want to do is to complete the square of the quadratic quantity underneath the square root sign. So, to complete the squared, we'll have X2 plus 6 X. Plus 10, and completing squared, this is going to be equal to the quantity of X plus 3 in quantity squared. Plus one. So here to complete the square, recall that you just need to take our linear term and divide it by 2, and this will give you the coefficient insight of the square term. So, half of 6 is equal to 3, so we have the quantity of X plus 3 and quantity squared. But in order to complete the square, we've essentially added 3 squad to this to this expression. And so we'll have to subtract 3 squad from the remaining constant. So we have 10 minus 32, which is 10 minus 9, which is going to give us 1. So now making this substitution into our expression for our integral, we see that our integral is going to be equal to the integral from minus 3 to 2 of 1 divided by the square root of the quantity of the quantity of X plus 3 in quantity squared plus 1 in quantity DX. And so now we can evaluate this integral by using U substitution. So, we'll set U equal to X + 3. So that means that DU is going to be equal to DX. So, with this substitution, we see that our integral is going to be equal to the integral from, and here we have to change our limits of integration. So when X is equal to minus 3 for the lower bound, we see that U is going to be equal to -3 + 3, which is going to be 0. And for the upper bound, when X is equal to 2, we see that U is going to be equal to 2 + 3, which is 5. And then in the integram we'll have 1 divided by the square root of the quantity of U2 plus 1 in quantity DU. And this is actually in a form that we can integrate. So recall that this integral is going to be equal to the inverse hyperbolic sign of you. And this is going to be evaluated from 0 to 5. However, we can rewrite the inverse hyperbolic sign function in terms of the natural log. So this is also equal to the natural log of the quantity of u plus the square root of the quantity of U2 + 1 in quantity evaluated at 0 and 5. And so, substituting in our limits of integration, we see that this is going to be equal to the natural log of the quantity of 5. Plus the square root of the quantity of 5 squad plus 1 in quantity, then we'll have minus for a lower limit, the natural log of the quantity of 0, plus the square root of the quantity of 0 squared plus 1 in quantity. And so simplify this expression, this is going to be equal to the natural log of the quantity of 5, plus the square root of 26. In quantity minus the natural log of 1. However, the natural log of 1 is 0, so this means that this is going to be equal to the natural log of the quantity of 5, plus the square root of 26 in quantity. So this here is the answer that we were looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

41–48. Geometry problems Use a table of integrals to solve the following problems.
46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.

Key Concepts

Definite Integrals

Area Under a Curve

Integral Tables

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