the instructions for the given integral have two parts, one for the trapezoidal rule and one for simpsons…

the instructions for the given integral have two parts, one for the trapezoidal rule and one for simpsons rule. complete the following parts.\n\n$$\\int_{-1}^{1}(2 x^{2}+1) d x$$\n\n a. estimate the integral with ( n = 4 ) steps and find an upper bound for ( |e_{t}| ).\n\n( t=\frac{7}{2} )\n(type an exact answer. type an integer or a simplified fraction.)\n\nthe upper bound for ( |e_{t}| ) is 0.17.\n(round to two decimal places as needed.)\n\n b. evaluate the integral directly and find ( |e_{t}| ).\n\n$$\\int_{-1}^{1}(2 x^{2}+1) d x=\\square$$\n(type an exact answer. type an integer or a simplified fraction.)

the instructions for the given integral have two parts, one for the trapezoidal rule and one for simpsons rule. complete the following parts.\n\n$$\\int_{-1}^{1}(2 x^{2}+1) d x$$\n\n a. estimate the integral with ( n = 4 ) steps and find an upper bound for ( |e_{t}| ).\n\n( t=\frac{7}{2} )\n(type an exact answer. type an integer or a simplified fraction.)\n\nthe upper bound for ( |e_{t}| ) is 0.17.\n(round to two decimal places as needed.)\n\n b. evaluate the integral directly and find ( |e_{t}| ).\n\n$$\\int_{-1}^{1}(2 x^{2}+1) d x=\\square$$\n(type an exact answer. type an integer or a simplified fraction.)

Answer

Explanation:

Step1: Find the antiderivative

The antiderivative of (2x^{2}+1) is (\frac{2}{3}x^{3}+x).

Step2: Apply the fundamental theorem of calculus

[ \begin{align*} \int_{-1}^{1}(2x^{2}+1)dx&=\left[\frac{2}{3}x^{3}+x\right]_{-1}^{1}\ &=\left(\frac{2}{3}(1)^{3}+1\right)-\left(\frac{2}{3}(-1)^{3}+(-1)\right)\ &=\left(\frac{2}{3}+1\right)-\left(-\frac{2}{3}-1\right)\ &=\frac{2 + 3}{3}-\left(\frac{-2-3}{3}\right)\ &=\frac{5}{3}-\left(-\frac{5}{3}\right)\ &=\frac{5}{3}+\frac{5}{3}\ &=\frac{10}{3} \end{align*} ]

Answer:

(\frac{10}{3})