Let the area of the bounded region {(x,y):0≤9x≤y2,y≥3x−6} be A. Then 6A is equal to _____
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Step-by-Step Solution
Step 1: Identify the curves and intersection points
The region is bounded by two curves: a parabola x=9y2 and a straight line x=3y+6. To find the area of the region, we first need to determine the points where these two curves intersect. We can do this by setting their x values equal to each other.
Step 2: Find the intersection points
To find the intersection points, we equate the expressions for x. This gives us a quadratic equation in y. Solving this quadratic equation by factoring, we find two values for y: y=6 and y=−3. These are the y -coordinates of the intersection points.
Step 3: Determine the x-coordinates of intersection
Now we substitute the y -values back into either of the original equations to find the corresponding x -coordinates. For y=6, x=4, giving the point (4,6). For y=−3, x=1, giving the point (1,−3). These are the two points where the line and the parabola intersect.
Step 4: Set up the integral for the area
Since the line x=3y+6 is to the right of the parabola x=9y2 in the region of interest, we integrate the difference of the x -values with respect to y. The limits of integration are the y -coordinates of the intersection points, from −3 to 6.
Step 5: Evaluate the integral
We now evaluate the definite integral. First, we find the antiderivative of each term. Then, we substitute the upper limit y=6 and the lower limit y=−3 into the antiderivative and subtract the results. This calculation yields the area A=227.
Step 6: Calculate 6A
The problem asks for the value of 6A. We multiply the calculated area A=227 by 6 to get the final answer.