Q. 1 - Calculus

Question

Functions Defined by Integrals: Suppose $A\left(x\right)$ is the function that for each $x>0$ is equal to the area under the graph of $f\left(t\right)=t^2$ from 0 to x. 

Write an equation that defines $A\left(x\right)$ in terms of a definite integral. (Hint: Let the variable inside the integral be t, so as not to confuse it with the variable x.) 

Step-by-Step Solution

Answer

an equation $A\left(x\right)$ defines in terms of a definate integral is   ${\int }_0^x f\left(t\right) dt=\frac{x^3}{3}$

Step 1. Given information

The area under the graph of $f\left(t\right)=t^2$

Step 2. Finding an equation that defines A ( x ) in terms of a definate integral

Suppose $A\left(x\right)$ is the function that for each $x>0$ is equal to the area under the graph of from 0 to x

The area under the graph of $f\left(t\right)=t^2$ can be written in the terms of definate integral as below

$$\begin{gathered} Area ={\int }_0^x f\left(t\right) dt \\ ={\int }_0^x t^2 dt \\ ={\left[\frac{t^3}{3}\right]}_0^x \\ =\frac{x^3}{3} \end{gathered}$$

Thus, an equation $A\left(x\right)$ defines in terms of a definate integral is ${\int }_0^x f\left(t\right) dt=\frac{x^3}{3}$