Area supplies another coefficient equation

The diagram below shows a portion of a quadratic curve y=f(x)y=f(x). The area of the shaded region bounded by the axes, the curve y=f(x)y=f(x) and the line x=1x=1 is 196\frac{19}{6} units2^2. Given also that the curve passes through the points (1,132)(1,\frac{13}{2}) and (−2,12)(-2,\frac12), determine the equation of the curve. marks

Quadratic y=f(x), shaded integral under exact curve between x=0 and x=1O1xyy = f(x)
Question diagram 2
Quadratic y=f(x), shaded integral under exact curve between x=0 and x=1O1xyy = f(x)

Native reconstruction of shaded region under the quadratic between x=0 and x=1

[2011/MJC/Prelim/I/6b]

Solution

Q

Integrals as constraints

A portion of a quadratic curve y=f(x)y=f(x) is shown. The area of the shaded region bounded by the axes, the curve and the line x=1x=1 is 196\frac{19}{6} units 2^2. The curve passes through (1,132)\left(1,\frac{13}{2}\right) and (−2,12)\left(-2,\frac12\right). Determine the equation of the curve.

Explore this idea

Hint

  1. Let f(x)=ax2+bx+cf(x)=ax^2+bx+c.

  2. Two points on the curve give two equations.

  3. The shaded area is ∫01f(x) dx\int_0^1f(x)\,dx: that is the third equation.

Answer · Q

y=2x2+4x+1/2y=2x^2+4x+1/2.

Explore the idea

The area under the fitted curve is 19/6

Strips under y = 2x² + 4x + ½ from x = 0 to x = 1 add up to 19/6 ≈ 3.167.

Number of stripsn = 4
Try an example

Where does one strip’s area come from?

1 of 4

Click a strip in the graph to inspect it. The darker rectangle is this strip. Its red marks use the left edge at x* = 0.

upperlowerwidth Δx
Zoomed-in rectangle: width and height are scaled independently. The sample sets its upper and lower values across the whole strip.
Sample x*
0
Width Δx
0.25
Upper value
0.5
Lower value
0
Height: upper − lower
0.5
Strip area
0.125

0.25 ×(0.5 − 0)≈ 0.125

Add this contribution to those of the other 3 strips to get the sum below. Displayed values are rounded.

Strip width
0.250
1.000 ÷ 4
Sum of strips
2.43750
Exact, 19/6
3.16667
Error
−0.72917
23.03% too small

How the sum improves as n grows

  • Fitted quadratic
  • Each of width 1/n
  • Given area 19/6
Key idea

The numerical area confirms the integral condition: the fitted quadratic really does enclose 196\frac{19}6 with the axes.

Ideas to try
With the right-edge rule and n = 40, how close is the sum to 19/6?

Three conditions on (a, b, c)

Two point conditions and one area condition are three planes in (a, b, c) space.

Open the interactive view to explore this geometry in 3D.