Finding the area under the curve of a function is called the **area problem**. Geometric shapes like squares and rectangles, which have defined formulas, can be used to find curved areas which *don’t* have common formulas.

The area problem only works for continuous functions that are non-negative (i.e. above the x-axis).

## The Area Problem: Example

**Example question:** What is the area of the shape under the curve of f(x) = x^{2} on the closed interval [1, 5]?

Step 1: **Graph the function** (I used Desmos.com):

Step 2: **Shade the area**.

We can see the “problem” here (and why it’s called *the area problem*); We don’t have an obvious formula for the shape created in the interval [1,5]. We could do some fandangling with an inverted ellipse (the shape looks like one quarter of an inverted one) but a *much easier way *is to approximate the area with rectangles.

Step 3: **Draw the rectangles**. We could draw any number of rectangles we like. For this example, we can get a reasonable approximation of the area (good enough for ordering tiles or carpet!) with 4 rectangles:

Step 4: **Calculate the area of each rectangle**. Calculate the area of each rectangle. There are two ways to do this: you could just look at the graph and ballpark the length and width of each rectangle. A better way is to use the function we’re given: x^{2}:

Rectangle # | Length ( on x-axis) | Height (on y-axis) | A = L * H |

1 | 1 | f(1.5) = (1.5)^{2} = 2.25 |
1 * 2.25 = 2.25 |

2 | 1 | f(2.5) = (2.5)^{2} = 6.25 |
1 * 6.25 = 6.25 |

3 | 1 | f(3.5) = (3.5)^{2} = 12.25 |
1 * 12.25 = 12.25 |

4 | 1 | f(4.5) = (4.5)^{2} = 20.25 |
1 * 20.25 = 20.25 |

Step 5: **Add up the rectangle areas from Step 4**.

2.25 + 6.25 + 12.25 + 20.25 = 41.

The area under the curve is approximately 41 units.

## Exact Solution

The solution of 41 is very close to the exact area (which we can get with a method called integration) of 124/3. To get the exact answer, we use definite integrals:

- Go to Symbolab’s Calculator.
- Click on the little grey boxes at the top and bottom of the integral symbol (∫) and type in 5 (At the top) and 1 (at the bottom). These are called integral bounds.
- Type in x
^{2}in between the integral symbol and the “dx”. Then click the red “Go” button on the right.

The exact solution is 124/3 ≈ 41.333.

## The Area Problem and Riemann Sums

In the above example, I used the midpoint of each interval to calculate the areas. I could also have used inscribed rectangles (under the curve) or circumscribed rectangles (above the curve). Calculation with all of these different sums are covered in the method of Riemann sums.