Q. 4 TF - Calculus

Question

Functions defined by area accumulation: We know that for fixed real numbers a and b and an integrable function f, the definite integral ${\int }_a^{b}f\left(x\right) dx$is a real number. For different real values of b, we get (potentially) different values for the integral ${\int }_a^{b}f\left(x\right) dx.$

What is the relationship between the formula that describes ${\int }_0^{b}2x dx$and the formula that describes ${\int }_1^{b}2x dx$

Step-by-Step Solution

Answer

 The relationship between the formula of both integrals is ${\int }_1^{b}2x dx=\left({\int }_1^{b}2x dx\right)-1.$

Step 1. Given information.

Given integral are ${\int }_a^{b}f\left(x\right) dx={\int }_0^{b}2x dx \& {\int }_a^{b}f\left(x\right) dx={\int }_a^{b}2x dx.$

Step 2. Relationship between the formula.

The formula for the integrals are following

$$\begin{gathered} {\int }_0^{b}2x dx=b^2 \cdots \left(i\right) \\ {\int }_1^{b}2x dx=b^2-1 \cdots \left(ii\right) \end{gathered}$$

Substitute ${\int }_0^{b}2x dx$ for $b^2$in formula ii.

$$\begin{gathered} {\int }_1^{b}2x dx=b^2-1 \\ {\int }_1^{b}2x dx=\left({\int }_1^{b}2x dx\right)-1 \end{gathered}$$