Find the area of the region bounded by the curves y2=4x and x2=4y.
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Step-by-Step Solution
Step 1: Identify the curves and find intersection points
We are given two parabolic equations. To find the area bounded by them, we first need to determine their points of intersection. We will solve these equations simultaneously to find the (x, y) coordinates where they intersect.
Step 2: Substitute y from (2) into (1)
From equation (2), we can express y as y=4x2. Substitute this expression for y into equation (1) to eliminate y and obtain an equation solely in terms of x.
Step 3: Solve for x-coordinates of intersection
Simplify the equation to solve for x. Factor out x from the equation x4−64x=0. This gives us two possibilities for x: x=0 or x3−64=0.
Step 4: Find the values of x
From x(x3−64)=0, we get x=0 as one solution. For the second solution, we solve x3=64, which yields x=4 since 43=64. These are the x -coordinates of the intersection points.
Step 5: Find the y-coordinates of intersection
Substitute the x -values back into either equation (1) or (2) to find the corresponding y -coordinates. Using y=4x2 (from equation 2), we find that when x=0, y=0, and when x=4, y=4. Thus, the curves intersect at (0,0) and (4,4).
Step 6: Set up the integral for the area
To find the area enclosed by the two curves, we integrate the difference between the upper curve and the lower curve with respect to x, from the first intersection point to the second. In this region, y2=4x means y=2x (upper half) and x2=4y means y=4x2 (lower curve). We need to determine which curve is 'upper' and which is 'lower' in the interval [0,4].
Step 7: Determine upper and lower curves
From y2=4x, we get y=±2x. Since the region of interest is in the first quadrant (between x=0 and x=4), we consider the positive root, y1=2x. From x2=4y, we get y2=4x2. We observe that for x∈(0,4), 2x>4x2. For example, at x=1, 21=2 and 412=0.25. So, y1=2x is the upper curve and y2=4x2 is the lower curve.
Step 8: Integrate to find the area
Now we set up the definite integral with the limits of integration from x=0 to x=4 and the integrand as the difference between the upper curve y=2x and the lower curve y=4x2.
Step 9: Perform the integration
Integrate each term with respect to x. Recall that ∫xndx=n+1xn+1. So, ∫xdx=∫x1/2dx=3/2x3/2 and ∫x2dx=3x3.
Step 10: Simplify the integrated terms
Simplify the constants. 2⋅3/21=2⋅32=34 and 41⋅31=121.
Step 11: Evaluate the definite integral
Apply the limits of integration by substituting the upper limit (x=4) and the lower limit (x=0) into the integrated expression and subtracting the result for the lower limit from the result for the upper limit.
Step 12: Calculate the numerical value
Calculate the values: 43/2=(4)3=23=8. Simplify 1264 by dividing both numerator and denominator by 4 to get 316. Subtract the terms.
Step 13: Final calculation
Perform the final subtraction to get the area.