A student attempts to evaluate the integral and writes the following solution: “

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

A student attempts to evaluate the integral

I=\displaystyle\int_{1}^{3}2x\,dx

and writes the following solution: “

\int 2x\,dx = x^2 + C

, therefore

I=[x^2+C]_{1}^{3}=(9+C)-(1+C)=8

. Since

C

is arbitrary, we take

C=0

and obtain

I=8

.” Study the work shown and answer the following questions. (i) Identify the conceptual error in the student’s method. (ii) Re-work the integral correctly to obtain the exact value of

I

. (iii) Explain clearly why the constant of integration

C

should not be introduced when evaluating a definite integral.

Related theory

Accumulation Concepts
Fundamental theorem of calculus
Definite integrals
Integrals

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