Approximating Definite Integrals — Question 1

Question 1

Given $$f(0)=1,\ f(1)=2,\ f(2)=5,\ f(3)=10,\ f(4)=17,$$ use $T_4$ to approximate $\int_0^4f(x)\,dx$.

Question 1 – Solution

Step 1: Find the subinterval width. $$h=\frac{b-a}{n}=\frac{4-0}{4}=1.$$ Step 2: Write the trapezoidal formula. $$T_4=\frac h2\left[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+f(x_4)\right].$$ Step 3: Substitute the table values. $$\begin{align*} T_4&=\frac12[1+2(2)+2(5)+2(10)+17]\\ &=\frac12[1+4+10+20+17]\\ &=\frac12(52)=26. \end{align*}$$ $$\boxed{T_4=26}$$