Determine \[ \int_0^{10}x^4-10x^3+5000\,dx. \]

Primary tabs

Lesson Parent: 
Chapter 27 - The Fundamental Theorem of Integral Calculus
Problem: 

Determine
\[
\int_0^{10}x^4-10x^3+5000\,dx.
\]

Answer: 

$\int_0^{10}x^4-10x^3+5000\,dx=45000$

Solution: 

The Fundamental Theorem of Calculus states that
\[
\int_0^{10}x^4-10x^3+5000\,dx=F(10)-F(0)
\]
where $F(x)$ is the antiderivative of $f(x)=x^4-10x^3+5000$. We determined $F(x)$ here and found that
\[
F(x)=\frac{1}{5}x^5-\frac{10}{4}x^4+5000x+C.
\]

So now
\[
F(10)=45000+C
\]
and
\[
F(0)=C
\]
so that
\[
F(10)-F(0)=45000.
\]