8. Application of Integrals

What you will be able to do

• Say what elementary geometry can measure and where it stops, and name the boundary that forces integration
• State both of the chapter's area formulas and say which strip direction each one belongs to
• Say what a definite integral reports when the curve lies under the horizontal axis, and why that number is not the area
• Say why the whole circle cannot be handed to the area formula directly

What this chapter assumes you already know

• The definite integral from the previous chapter, both as a limit of sums and as an evaluated antiderivative
• The two theorems of that chapter that let a definite integral be computed by subtracting antiderivative values at the two limits
• Reading a graph: what it means for a point to lie on a curve, and what the height of a curve above a horizontal axis is
• The area formulas of school geometry for a rectangle, a triangle, a trapezium and a circle
• Function notation, and the idea that a curve given by a rule assigns one height to each input
• The four-quadrant coordinate plane, and what a vertical line with a fixed first coordinate is
• The strip construction of the previous topic: a region cut into parallel slices, one slice written as height times width, the total written as an integral
• Rearranging a curve's equation to make either variable the subject

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

• "The strips are actually rectangles, so the answer is approximate."
• "The picture proves the formula."
• "The two vertical lines are just there to show where to stop drawing."
• "Any thin shape will do as a strip."
• "The height of the strip is a fixed number."
• "An integral means an antiderivative."
• "The region has to sit against the vertical axis."
• "Area under a curve and area between a curve and the axis are different ideas."