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Calculus IIChapter 1: Integration TechniquesSection 1-10

Approximating Definite Integrals — Question 1

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Question 1

Given f(0)=1,f(1)=2,f(2)=5,f(3)=10,f(4)=17,f(0)=1,\ f(1)=2,\ f(2)=5,\ f(3)=10,\ f(4)=17, use T4T_4 to approximate ∫04f(x)dx\int_0^4f(x)\,dx.

Original worksheet page 1: question and worked solution for 1-10-001
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Question 1 – Solution

Step 1: Find the subinterval width. h=b−an=4−04=1.h=\frac{b-a}{n}=\frac{4-0}{4}=1. Step 2: Write the trapezoidal formula. T4=h2[f(x0)+2f(x1)+2f(x2)+2f(x3)+f(x4)].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. T4=12[1+2(2)+2(5)+2(10)+17]=12[1+4+10+20+17]=12(52)=26.\begin{align*} T_4&=\frac12[1+2(2)+2(5)+2(10)+17]\\ &=\frac12[1+4+10+20+17]\\ &=\frac12(52)=26. \end{align*} T4=26\boxed{T_4=26}

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