If the area of the region \{(x,y): |x-5| ≤y≤4x} is A, then 3A is equal to _____
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Step-by-Step Solution
Step 1: Identify the bounding curves
The region is bounded below by the function y1=∣x−5∣ and above by the function y2=4x. We need to find the area between these two curves.
Step 2: Find intersection points
To find the limits of integration, we need to determine where the two curves intersect. We set y1=y2 and solve for x. We consider two cases for ∣x−5∣: x−5 for x≥5 and −(x−5) for x<5.
Step 3: Solve for intersection points (Case 1: x≥5)
For x≥5, we have x−5=4x. Squaring both sides gives (x−5)2=(4x)2, which simplifies to x2−10x+25=16x. Rearranging gives x2−26x+25=0. Factoring this quadratic equation yields (x−1)(x−25)=0. Since we assumed x≥5, the valid solution is x=25.
Step 4: Solve for intersection points (Case 2: x<5)
For x<5, we have −(x−5)=4x, which is 5−x=4x. Squaring both sides gives (5−x)2=(4x)2, which simplifies to 25−10x+x2=16x. Rearranging gives x2−26x+25=0. Factoring this quadratic equation yields (x−1)(x−25)=0. Since we assumed x<5, the valid solution is x=1.
Step 5: Set up the integral for area A
The area A is found by integrating the difference between the upper curve y2=4x and the lower curve y1=∣x−5∣. Since ∣x−5∣ changes its definition at x=5, we split the integral into two parts: from x=1 to x=5 (where y1=5−x) and from x=5 to x=25 (where y1=x−5).
Step 6: Evaluate the integrals
We evaluate each integral separately. The antiderivative of 4x=4x1/2 is 4⋅32x3/2=38x3/2. The antiderivative of 5−x is 5x−2x2. The antiderivative of x−5 is 2x2−5x.
Step 7: Calculate the definite integrals and find A
Substitute the limits of integration into the antiderivatives and perform the arithmetic. After careful calculation, the total area A is found to be 3200.