Q. 4 - Calculus

Question

Use the Fundamental Theorem of Calculus to find an equation for A(x) that does not involve an integral.

Step-by-Step Solution

Answer

$A\left(x\right)\approx \sum _{}^{n}f\left(\frac{1}{x}\right)\frac{1}{x}$

Step 1. Given information is:

Suppose A(x) is a function that for each x>0 is equal to area under the graph of f(t)=t²  from 0 to x.

Step 2. Determining A(x)

$$\begin{gathered} The area under the curve is given by: \\ A\left(x\right)=f\left(\frac{1}{x}\right)\frac{1}{x}+f\left(\frac{1}{x}\right)\frac{1}{x}+.....+f\left(\frac{1}{x}\right)\frac{1}{x} \\ A\left(x\right)\approx \sum _{}^{n}f\left(\frac{1}{x}\right)\frac{1}{x} \\ Thus, A\left(x\right)\approx \sum _{}^{n}f\left(\frac{1}{x}\right)\frac{1}{x} \end{gathered}$$