Build the square or rectangle area

Squares on a curved region

A solid stands on the region between the x-axis and the curve \(y = 2\sqrt{x}\). Each cross section is perpendicular to the x-axis. The teal face is one cross section, and it is a square.

The side of the square is the orange base on the floor. At \(x = 1\), the curve is at height 2, so the side is 2 and the area is \(2 \cdot 2 = 4\).

Drag the slice to x = 4. The side is 4 and the area is 16, so both numbers depend on x.

Press the button. The face becomes a rectangle with the same base and half the height. At \(x = 4\), the base is 4 and the height is 2, so the area is \(4 \cdot 2 = 8\).

Base times height

The orange base has length w, and the orange upright side is the height h.

Drag the slider. The face is a rectangle, and its area is the base times the height. When the height equals the base, the face is a square.

The area of a square is \(w \cdot w = w^2\). The base depends on \(x\), so the area depends on \(x\).

Notation

1. A square has four equal sides, and the orange base is one side, so the area is the base width times itself.

   \[\cb{A(x)} = \cy{w(x)}^2\]

2. A rectangle has the orange base and an upright height \(\cy{h}\), which the problem states, so its area is the base times the height.

   \[\cb{A(x)} = \cy{w(x)} \cdot \cy{h}\]

3. The height can be a number, a multiple of the base, or a formula in \(x\). In every case, write \(A\) in terms of \(x\) only.

4. If \(w(x) = f(x) - g(x)\), square the whole difference, because the side of the square is the difference.

   \[\left(f(x) - g(x)\right)^2 \ne f(x)^2 - g(x)^2\]

Example 1: rectangles of a fixed height

The base of a solid is the region between the x-axis and \(y = 6 - 2x\), for \(0 \le x \le 3\). Its cross sections perpendicular to the x-axis are rectangles of height 5. Find \(A(x)\).

1. The lower boundary is the x-axis, so the base width is the height of the line.

   \[\cy{w(x)} = 6 - 2x\]

2. The problem gives the height of the rectangle as 5, the same for every value of x.

3. Multiply the base by the height.

   \[\cb{A(x)} = 5(6 - 2x) = 30 - 10x\]

4. As a check, take \(x = 1\). The base is 4 and the height is 5, so the area is 20. The formula gives \(30 - 10 = 20\).

Example 2: squares between two curves

The base of a solid is the region between \(y = x\) and \(y = x^2\), for \(0 \le x \le 1\). Its cross sections perpendicular to the x-axis are squares. Find \(A(x)\).

1. For \(0 \le x \le 1\), \(x \ge x^2\), so the line is the upper curve. The side of the square is the difference of the heights.

   \[\cy{w(x)} = x - x^2\]

2. A tempting wrong move: Square each curve. Then subtract.

   \[\cr{A(x) = x^2 - \left(x^2\right)^2 = x^2 - x^4}\]

   At \(x = \tfrac12\), the side is \(\tfrac12 - \tfrac14 = \tfrac14\), so the area is \(\tfrac{1}{16}\). This formula gives \(\tfrac{3}{16}\), the area of the red square, which is three times too large.

3. Subtract first. Then square the whole side.

   \[\begin{aligned} \cb{A(x)} &= \left(x - x^2\right)^2 \\ &= x^2 - 2x^3 + x^4 \end{aligned}\]

4. At \(x = \tfrac12\), this gives \(\tfrac{1}{16}\), the area of the teal square.

Practice

The base of a solid is the region between the x-axis and \(y = \frac{4}{x}\), for \(1 \le x \le 4\). Its cross sections perpendicular to the x-axis are rectangles of height \(x + 1\). Find \(A(x)\).

The area is the orange base times the upright height of the face.

Questions

The base of a solid lies between the parabola \(y = \frac{x^2}{2}\) and the line \(y = 2\). Its cross sections are perpendicular to the x-axis.

1. The slice at \(x = 1\) is a square. What is its area?

   • \(\frac{9}{4}\)
   • \(\frac{3}{2}\)
   • 3
   • \(\frac{15}{4}\)
   Answer

   \(\frac{9}{4}\)

2. The slice at \(x = 0\) is a rectangle. Its height is half its base. What is its area?

   • 2
   • 4
   • 3
   • 1
   Answer

   2

3. A base lies between \(y = x\) and \(y = 3x\). The cross sections are squares perpendicular to the x-axis. What is \(A(x)\)?

   • \(4x^2\)
   • \(2x\)
   • \(8x^2\)
   Answer

   \(4x^2\)

Summary

You write the orange base width as a difference of heights, and then you square it or multiply it by the stated height.

This works because the area of a square is the side times itself, and the area of a rectangle is the base times the height.

You use it to write \(A(x)\) before you set up a volume integral.

This is one of four skills in topic 8.7 of Which & Why: AP Calculus.