Chapter 7, Problem 3RWDT

To determine

To check: The area of R when f and g multiplied by the positive constant c . Also, to check what happens when only f is multiplied by c .

Answer to Problem 3RWDT

The new area will be c times the original area.

If only f is multiplied by positive constant c , then $A_{new}={\displaystyle \underset{a}{\overset{b}{\int }}\left(cf\left(x\right)-g\left(x\right)\right)}dx$ , where c is a positive constant.

Explanation of Solution

Given information:

The region R lies in the I quadrant and has upper boundary: $y=f\left(x\right),$

Left boundary: $x=a$ ,

Right boundary: $x=b$ , and

Lower boundary: $y=g\left(x\right)$ , where f and g are continuous functions.

Concept used:

Area between the curves $y=p\left(x\right)$ $\left(\text{bottom}\right)$ and $y=q\left(x\right)$ $\left(\text{top}\right)$ from $x=m$ to $x=n$ is given by:-

  $A={\displaystyle {\int }_m^n(q(}x)-p(x))dx$

Area between the curves $y=g\left(x\right)$ $\left(\text{bottom}\right)$ and $y=f\left(x\right)$ $\left(\text{top}\right)$ from $x=a$ to $x=b$ is given by:-

  $A={\displaystyle {\int }_a^b(f(}x)-g(x))dx$

When f and g multiplied by the positive constant c , both the curves get vertically stretched then the new area of the region R will be:

  $\begin{array}{c}{A}_{new}={\displaystyle \underset{a}{\overset{b}{\int }}\left(cf\left(x\right)-cg\left(x\right)\right)}dx\\ ={\displaystyle {\int }_a^{b}c\left(f(x)-g(x)\right)}dx\\ =c{\displaystyle \underset{a}{\overset{b}{\int }}\left(f(x)-g(x)\right)}dx\left[{\displaystyle \int cf\left(x\right)dx=c{\displaystyle \int f\left(x\right)dx}}\right]\\ =cA\\ A_{new}=cA\end{array}$

The new area will be c times the original area.

If only f is multiplied by positive constant c , then only f gets vertically stretched or compressed $\left(\text{based on the value of }c\right)$.

  $A_{new}={\displaystyle \underset{a}{\overset{b}{\int }}\left(cf\left(x\right)-g\left(x\right)\right)}dx$ , where c is a positive constant.