- Topics At a Glance
**Area**- Assumptions When Finding Area
**Triangles**- Circles
- Integrating with Polar Coordinates
- Volume
- Volumes of Solids with Known Cross-Sections
- Pyramids, Cones, and Spheres
- Volume of Solids of Revolution
- Disks and Washers
- Shells
- Washers vs. Shells
- Arc Length
- Arc Length for Parametric Functions
- Arc Length for Polar Functions
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

We've made it. We survived years of seemingly endless lectures about trigonometry, proofs, and functions. Your 7th grade math teacher rattled off formula after formula for the areas and volumes of different polygons and geometries. All we could do is sit there, listen, and memorize every formula we were given.

We were like power lines transporting the power—our geometry formula—from a power plant that produced the formula to the paper or project, the formulas final home. With calculus, we will never make a memory error about a length, area, or volume formula again. If we aren't sure, we can produce our own formula on the spot.

With our new found power, we will begin our production slowly. Triangles are the simplest and most useful polygons we come across, so we will *derive* the formula for the area of a triangle first.

Example 1

Find an integral expression for the area of the triangle, using slices as shown. Use |

Exercise 1

For each triangle,

(a) Find an integral expression for the area of the triangle using the slice and variable of integration indicated.

(b) Evaluate your integral and check that it agrees with the area you find using the formula

Exercise 2

For each triangle,

(a) Find an integral expression for the area of the triangle using the slice and variable of integration indicated.

(b) Evaluate your integral and check that it agrees with the area you find using the formula

Exercise 3

For each triangle,

(a) Find an integral expression for the area of the triangle using the slice and variable of integration indicated.

(b) Evaluate your integral and check that it agrees with the area you find using the formula

Exercise 4

For each triangle,

(b) Evaluate your integral and check that it agrees with the area you find using the formula

Exercise 5

For a triangle with base *b* and height *h*, use integration to derive the area formula

Use vertical slices with *x* for the variable of integration.