Flower Bed Area Between a Parabola and a Path
A landscaper is designing a flower bed in a public park. On the plan, one edge is the parabola y=x² and the other edge is the straight path y=2x+3. The bed lies between the two curves. A contractor measures the two crossing points at x=−1 and x=3, treats the bed as a triangle with base 4 m and height 9 m, and quotes for 18 m² of soil. The landscaper suspects this is wrong because the left-hand crossing is also part of the boundary and because the path is above the parabola on the whole bed. The park also wants a sprinkler line along the vertical line x=1; the question is whether that line passes through the bed and how the bed area is split by it. All lengths are in metres.
1. Find the intersection points of y=x² and y=2x+3, and explain why the straight path is the upper boundary on the whole bed.
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Set x²=2x+3, so x²−2x−3=0. Factor: (x−3)(x+1)=0, giving x=−1 and x=3. The corresponding y-values are y=(−1)²=1 and y=3²=9, so the crossing points are (−1,1) and (3,9). To decide which curve is above, examine the difference d(x)=(2x+3)−x². At x=0, d(0)=3>0, so the line is above the parabola between the crossings. Since d is a downward-opening quadratic with zeros only at −1 and 3, it stays positive on the open interval (−1,3). Therefore the straight path y=2x+3 is the upper boundary and y=x² is the lower boundary on the entire bed. The bed is not a triangle; its top is a straight line and its bottom is curved.
2. Compute the exact area of the bed and compare it with the contractor's 18 m² quote.
Read the worked answer
Area = ∫ from −1 to 3 of [(2x+3)−x²] dx. An antiderivative is x²+3x−x³/3. At x=3: 9+9−9=9. At x=−1: 1−3+1/3=−5/3. Therefore the area is 9−(−5/3)=9+5/3=32/3≈10.67 m². The contractor's triangle uses base 4 m and height 9 m, giving (1/2)(4)(9)=18 m². The quote is too high by 18−32/3=(54−32)/3=22/3≈7.33 m², which is more than double the true area. The triangular model overfills the curved region because it includes area above the straight path near the ends and ignores the curvature of the lower boundary.
3. The sprinkler line x=1 divides the bed. Find the area on each side and state which side is larger.
Read the worked answer
Left part: ∫ from −1 to 1 of (2x+3−x²) dx = [x²+3x−x³/3] from −1 to 1. At x=1: 1+3−1/3=11/3. At x=−1: −5/3. Difference = 11/3−(−5/3)=16/3≈5.33 m². Right part: total 32/3 minus 16/3 = 16/3≈5.33 m². The two sides are exactly equal. This is not an accident: d(x)=2x+3−x² is a downward parabola with axis x=1, so the height function is symmetric about the sprinkler line. The integration confirms that symmetry.