Question

The area (A) under a curve is equal to the sum of the areas of n rectangles, taking the limit as n approaches infinity. Which is the correct mathematical representation of the area?What's the answer? A,BC, or D?

Answer 1

The area of each k-th rectangle is

$\begin{gathered} f(x_k)\cdot\Delta x_{} \\ \\ \text{where }f(x_k)\text{ is the height of rectangle k, and }\Delta x\text{ is the side of each rectangle} \end{gathered}$

Thus, after summing the areas of the n rectangles, we obtain:

$\sum ^n_{k=1}(f(x_k)\cdot\Delta x_{})$

Then, to find the area under the curve we need to take the limit as n approaches infinity. So, we obtain:

$\lim _{n\to\infty}\sum ^n_{k=1}(f(x_k)\cdot\Delta x_{})$

Therefore, option A is correct.