Approximating Definite Integrals — Question 3

Question 3 – Solution

Step 1: Find the width and nodes. $$h=\frac{2-0}{4}=0.5,\qquad x_i=0,0.5,1,1.5,2.$$ Step 2: Evaluate $f(x)=\sqrt{1+x^3}$. $$\begin{align*} f(0)&=1,& f(0.5)&=\sqrt{1.125},& f(1)&=\sqrt2,\\ f(1.5)&=\sqrt{4.375},& f(2)&=3. \end{align*}$$ Step 3: Apply Simpson’s Rule. $$\begin{align*} S_4&=\frac h3[f(0)+4f(0.5)+2f(1)+4f(1.5)+f(2)]\\ &=\frac{0.5}{3}\left[1+4\sqrt{1.125}+2\sqrt2 +4\sqrt{4.375}+3\right]\\ &\approx3.23961135. \end{align*}$$ $$\boxed{S_4\approx3.2396}$$