id: 06560915
dt: j
an: 2016c.00855
au: Barragués, José I.; Morais, Adolfo
ti: A classroom note to help student understanding of the accumulation
function, $F(x)=\int_a^x y(t)dt$.
so: Math. Comput. Educ. 49, No. 3, 194-204 (2015).
py: 2015
pu: MATYC Journal, Old Bethpage, NY
la: EN
cc: I55
ut: university teaching; integral calculus; concept formation; student
activities; concepts; accumulation function; average value; definite
integrals; Barrow’s rule; fundamental theorem of calculus;
mathematical applications; mathematical model building
ci:
li:
ab: From the introduction: For many students the concept of the integral is
reduced to the search of inverse derivatives and its interpretation as
the area under a curve. Students may also identify the integral with
the rule of Barrow, even when such rule cannot be applied. For example,
many students are unable to explain why it is incorrect to apply
Barrow’s rule to the function $f(x)=x-2$ in the interval $[-1,1]$,
obtaining: $\int_{-1}^{1} \frac{1}{x^2} dx=-2$. Students may know
different methods of integration, but may not be able to apply them
appropriately. In addition, a large number of students may identify
“integral" with “antiderivative", without being aware of any
process of convergence. The concept of accumulation is central to the
idea of integration, and therefore is at the core of understanding many
ideas and applications in Calculus. Analysis of the accumulation
function, $F(x)$, has great theoretical importance because it paves the
way to the Fundamental Theorem of Calculus. Nevertheless, the
definition of $F(x)$ can seem very artificial to students. We present
the following activities to help first-year college and university
students better understand the concept of integration.
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