Functions defined by area accumulation: We know that for fixed real numbers a and b and an integrable functionf, 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 ${\int }_1^{b}2xdx$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.

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