The area of the region, inside the circle (x−23)2+y2=12 and outside the parabola y2=23x is:
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Step-by-Step Solution
Step 1: Identify the equations and find intersection points
We are given the equations of a circle and a parabola. The circle is centered at (23,0) with radius 12=23. The parabola opens to the right with its vertex at the origin. To find the area of the region, we first need to find the points where the circle and parabola intersect.
Step 2: Solve for intersection points
Substitute the equation of the parabola, y2=23x, into the equation of the circle. This eliminates y and allows us to solve for the x -coordinates of the intersection points. We expand the circle equation and simplify to get a quadratic equation in x.
Step 3: Determine intersection coordinates
From the simplified quadratic equation, we find two possible values for x: x=0 and x=23. Substitute these values back into the parabola equation y2=23x to find the corresponding y -coordinates. This gives us the intersection points (0,0), (23,23), and (23,−23).
Step 4: Set up the integral for the area
The required area is the region inside the circle and outside the parabola. Due to symmetry about the x -axis, we can calculate the area in the upper half and multiply by 2. The upper boundary is given by the circle y=12−(x−23)2 and the lower boundary by the parabola y=23x. The integration limits are from x=0 to x=23.
Step 5: Evaluate the integral for the circular segment
Let u=x−23, so du=dx. When x=0, u=−23. When x=23, u=0. The integral becomes ∫−23012−u2du. This represents the area of a quarter circle of radius 12=23. Thus, the integral evaluates to 41π(23)2=41π(12)=3π.
Step 6: Evaluate the integral for the parabolic segment
Now we evaluate the integral for the parabolic part. We integrate x1/2 and apply the limits of integration. This gives us 23⋅32(23)3/2=32(23)2=32(12)=8.
Step 7: Calculate the total area
The total area is twice the difference between the area under the circular arc and the area under the parabolic arc from x=0 to x=23. Substituting the calculated values, we get 2(3π−8)=6π−16.