Approximating Definite Integrals — Question 2

Question 2

Use $M_4$ to approximate $$\int_0^2e^{-x^2}\,dx.$$ Give the exponential form and a decimal.

Question 2 – Solution

Step 1: Find the width. $$h=\frac{2-0}{4}=0.5.$$ Step 2: Find the four midpoints. $$0.25,\qquad0.75,\qquad1.25,\qquad1.75.$$ Step 3: Apply the midpoint formula. $$\begin{align*} M_4&=h\sum_{i=1}^4f(m_i)\\ &=0.5\left(e^{-(0.25)^2}+e^{-(0.75)^2} +e^{-(1.25)^2}+e^{-(1.75)^2}\right). \end{align*}$$ Step 4: Evaluate. $$M_4\approx0.88278895.$$ $$\boxed{M_4\approx0.8828}$$