Q. 3 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 ${\int }_a^{b}f\left(x\right) dx.$

Now make a table of the values of the integral ${\int }_1^{b}2x dx$corresponding to the values $-3,-2,-1,0,1,2, \mathrm{and} 3$for b. Conjecture a formula for the relationship between the values of b and the corresponding value of the integral. 

Step-by-Step Solution

Answer

The formula for the integral is ${\int }_1^{b}2x dx=b^2-1$and the table of values of integral for different values of b is following.

Step 1. Given information.

The given integral is ${\int }_a^{b}f\left(x\right) dx={\int }_1^{b}2x dx.$

Given values of b are $-3,-2,-1,0,1,2, \mathrm{and} 3.$

Step 2. Integral of ∫ 1 b 2 x   d x .

Determine the integral of ${\int }_1^{b}2x dx.$

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

So ${\int }_1^{b}2x dx=b^2-1.$

Step 2. Values of integral.

Substitute $-3,-2,-1,0,1,2, \mathrm{and} 3$for b in the integral.