Topic 8.7 Notes – Volumes with Cross Sections: Squares and Rectangles

Volume from Known Cross Sections

Imagine slicing a solid into infinitely thin pieces. Each slice has:

• Area $A(\text{variable})$
• Thickness $dx$ or $dy$

Add them up with a definite integral:

$$V = \int_a^b A(x)\,dx \quad \text{or} \quad V = \int_c^d A(y)\,dy$$

What matters:

• $A(\text{variable})$ is the area formula of the cross section
• The variable matches the slice direction
• The bounds come from the base region

Think of it as area-between-curves, but each “height” is now the area of a shape.

If slices are:

• Perpendicular to the x-axis → integrate with respect to $x$
• Perpendicular to the y-axis → integrate with respect to $y$

Types of Cross Sections on the AP Exam

Square Cross Sections

Area of a square:

$$A = s^2$$

The side length $s$ comes from the base region.

If using vertical slices:

$$s = \text{top function} - \text{bottom function}$$

If using horizontal slices:

$$s = \text{right function} - \text{left function}$$

So volume becomes:

$$V = \int (\text{distance between curves})^2 \, d(\text{variable})$$

The vertical segment represents the side of the square cross section.

One common mistake on quizzes is forgetting the parentheses:

$$(f - g)^2 \neq f^2 - g^2$$

Rectangular Cross Sections

Area of a rectangle:

$$A = w \cdot h$$

Usually:

• One dimension comes from the distance between curves
• The other is:
  • A constant
  • A multiple of the base
  • Another function

Example structure:

• Base region between $y=\sqrt{x}$ and $y=0$
• Rectangles perpendicular to the x-axis
• Height is 3 times the base

Then:

• Base $= \sqrt{x} - 0 = \sqrt{x}$
• Height $= 3\sqrt{x}$
• Area $= (\sqrt{x})(3\sqrt{x}) = 3x$

Volume:

$$V = \int_a^b 3x \, dx$$

Always identify which dimension comes from the region and which is given.

How to Set Up the Integral

When you see one of these on a test, the setup is everything.

1. Sketch the base region

   

   Region bounded by $x = y^2$ and $x = 4$ with a horizontal slice (right − left)

   Even rough sketches prevent direction mistakes. Notice how the slice is horizontal, so the distance is measured right − left.

2. Find bounds

   • Solve intersections, or
   • Use given boundary lines

3. Match slice direction to variable

   • Perpendicular to x-axis → $dx$
   • Perpendicular to y-axis → $dy$

4. Write the distance correctly

   • Vertical slices → top − bottom
   • Horizontal slices → right − left
     (If integrating in $y$, rewrite equations as $x =$ something.)

5. Plug into the area formula

   • Square → distance²
   • Rectangle → width × height

Then integrate and evaluate.

Switching Between x and y

This is where students lose easy points.

If integrating with respect to $y$:

• You measure horizontal distance
• Use right − left
• Rewrite equations as $x = f(y)$

If integrating with respect to $x$:

• You measure vertical distance
• Use top − bottom

Your slice direction, distance expression, and variable must all agree. If one doesn’t match, the setup is wrong.