CalcNotes The Fundamental Theorem of Calculus

Section 5.4 The Fundamental Theorem of Calculus

If \(f\) is continuous on \([a,b]\text{,}\) then we can define

\begin{equation*} F(x) = \int_a^x f(t)\, dt \end{equation*}

Then \(F\) is continuous on \([a,b]\text{,}\) differentiable on \((a,b)\text{,}\) and

\begin{equation*} F'(x) = \frac{d}{dx} \int_a^x f(t)\, dt = f(x) \end{equation*}

If \(f\) is continuous on \([a,b]\) and \(F\) is any antiderivative of \(f\text{,}\) then

\begin{equation*} \int_a^b f(x)\, dx = F(b)-F(a) \end{equation*}

which is often written as

\begin{equation*} \int_a^b f(x)\, dx = \left.F(x)\right|_a^b \end{equation*}

Compute:

\(\displaystyle \displaystyle\int_2^5 x^3 + 4x + 2\, dx\)
\(\displaystyle \displaystyle\int_0^{\pi/3} \sec\theta \tan\theta\, d\theta\)
\(\displaystyle \displaystyle\int_4^{-4} (3r+2)^2\, dr\)
\(\displaystyle \displaystyle\int_{-3}^{-2} \frac{y^6-5y^2}{y^5}\, dy\)
\(\displaystyle \displaystyle\int_{-5}^3 |x|\, dx\)
\(\displaystyle \displaystyle\int_0^{7\sqrt{3}} \frac{19}{\sqrt{196-s^2}}\, ds\)

Find the total area of the region between the \(x\)-axis and \(y=x^{1/3}-x^3\) over \(-1 \leq x \leq \sqrt[4]{8}\text{.}\)

\(2\)