Determine the area of the shaded region in the following figur...

Question

Determine the area of the shaded region in the following figures.

Graph showing two curves with a shaded area between them, labeled with equations for each curve.

Accepted Answer

Welcome back, everyone. Find the area enclosed by the shaded region and the figure below. A 2 minus 4/3, B 2 minus 2/3, C 2 + 4/3, and D 4/3 minus 2. For this problem, let's recall that whenever we want to identify the area enclosed by two curves, we have to integrate a difference between Two functions, right? First of all, we have to identify the upper curve and as the image suggests, we have a red curve, Y equals 4 divided by X2 + 1 as our upper curve, right? It is above the blue one. And then we have The lower curve, which is the blue one, that's our parabola. Y equals to x2. So we take the upper curve. Which is 4 divided by X2 + 1, we subtract the lower curve, which is 2 X2, and we want to integrate this difference with respect to X. Now, the limits of integration will be found by the intersection points. First of all, let's notice that our first intersection point is at X equals -1, so that'll be the lower limit of integration. And the upper limit of integration is the 2nd intersection point at X equals 1. That that's our upper limit. Well then, so now we have our integral. We can also simplify it using symmetry. Let's notice that this region is symmetric with respect to the y axis, so we can rewrite it as 2 integral from 0 to 1 of 4 divided by X2 + 1 minus 2 X 2 D X. In other words, we're finding the area between -1 and 0 and multiplying it by 2 to take into account the right side that is symmetric to the left side of the total region. Now let's go ahead and evaluate the result. We are going to factor out the constant, which is 2. Now let's integrate 4 divided by X2 + 1. We can factor out 4, and the integral of 1 divided by X2 + 1 D X is inverse of tangent of X. And then we can factor out -2 for the second part and integrate X2d. According to the power rule, that's X cubed divided by 3. Well then, so we have our integrals and now we're evaluating the result from 0 to 1. Let's substitute 1 for every X, we get 2 re, 4 inverse of tangents, of 1 minus 2 multiplied by 1 cubed, divided by 3. And then we subtract the value of this whole function adds 0, which is basically 0 inverse of tangent of 0 is 0. And 2/3 multiplied by 0 cubed is 0 as well. This is why symmetry is useful here. We essentially can simplify the calculation, right? This is equal to 2 multiplied by. In parencies we get 4 multiplied by inverse of tangent of 1, which is pi divided by 4 minus 2/3. Distributing, we get 2 multiplied by pi, which is 2 minus 2 multiplied by 2/3, which is minus 4/3. So the correct answer to this problem is a 2 pi minus 4/3. Thank you for watching.

Determine the area of the shaded region in the following figures.

Key Concepts

Definite Integral

Finding Points of Intersection

Area Between Curves

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