How do we compute the area of the first rectangle after partitioning?

Conditions

• The interval $[1, 2]$ is divided into $n$ equal subintervals.
• The height is evaluated at the right endpoint of the subinterval.

Reasoning, step by step

1. Identify the width of the first subinterval, which is $\frac{1}{n}$.
2. Determine the x-coordinate of the right endpoint of the first subinterval: $x = 1 + \frac{1}{n}$.
3. Evaluate the function $y = \frac{1}{x}$ at this x-coordinate to find the height: $\frac{1}{1 + \frac{1}{n}}$.
4. Multiply the width by the height to get the area of the first rectangle.

Example

The video states: 'The area of the first rectangle... its base is one over n, multiplied by the function value at this point... the x-coordinate of this point is 1 plus one over n... substitute this x-coordinate into the function expression... yielding this expression.' The formula shown is $\frac{1}{n} \cdot \frac{1}{1+\frac{1}{n}}$.

Common misconceptions

• Using the left endpoint instead of the right endpoint for the height.
• Forgetting to multiply by the width $\frac{1}{n}$ and only calculating the function value.

Watch the explanation

Explore next

• How do we generalize this to find the area of the k-th rectangle?
• What is the total area of all n rectangles?