Determine \[ \int_0^ax\,dx. \]
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Lesson Parent:
Problem:
Determine
\[
\int_0^ax\,dx.
\]
Answer:
$\int_0^ax\,dx=a^2/2$
Solution:
The Fundamental Theorem of Calculus states that
\[
\int_0^ax\,dx=F(a)-F(0)
\]
where $F(x)$ is the antiderivative of $f(x)=x$. We determined $F(x)$ here and found that
\[
F(x)=\frac{1}{2}x^2+C.
\]
So now
\[
F(a)=\frac{1}{2}a^2+C
\]
and
\[
F(0)=C
\]
so that
\[
F(a)-F(0)=\frac{1}{2}a^2.
\]
We could have determined this result by knowing that we are just looking for the area of the triangle under $f(x)=x$ between $0$ and $a$. Because this is one half the area of a square with sides of length $a$, we know the area is $a^2/2$. See the picture below.
