Question

estimate $int_{0}^{1}cos(x^{2})dx$ using the trapezoidal rule with $n = 4$. from a graph of the integrand, decide whether your answer is an underestimate or overestimate of the integral.

Explanation:

Step1: Calculate $\Delta x$

The formula for $\Delta x=\frac{b - a}{n}$, where $a = 0$, $b = 1$, and $n=4$. So $\Delta x=\frac{1 - 0}{4}=\frac{1}{4}=0.25$.

Step2: Find the $x$-values

The $x$-values are $x_0 = 0,x_1=0.25,x_2 = 0.5,x_3=0.75,x_4 = 1$.

Step3: Evaluate the function at $x$-values

$f(x)=\cos(x^{2})$, so $f(x_0)=\cos(0^{2}) = 1$, $f(x_1)=\cos(0.25^{2})\approx\cos(0.0625)\approx0.998$, $f(x_2)=\cos(0.5^{2})=\cos(0.25)\approx0.969$, $f(x_3)=\cos(0.75^{2})=\cos(0.5625)\approx0.845$, $f(x_4)=\cos(1^{2})\approx0.540$.

Step4: Apply the Trapezoidal Rule

The Trapezoidal Rule formula is $T_n=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+f(x_4)]$.
Substitute the values: $T_4=\frac{0.25}{2}[1 + 2\times0.998+2\times0.969+2\times0.845+0.540]$.
First, calculate the sum inside the brackets: $1+1.996 + 1.938+1.69+0.540=7.164$.
Then, $T_4=\frac{0.25}{2}\times7.164=0.8955$.

Step5: Determine if it's an over - or under - estimate

The second - derivative of $y = f(x)=\cos(x^{2})$ is $y''=-2\cos(x^{2})+4x^{2}\sin(x^{2})$.
For $x\in[0,1]$, $y''<0$, so the function $y = \cos(x^{2})$ is concave - down on $[0,1]$. When using the Trapezoidal Rule on a concave - down function, the result is an underestimate.

Answer:

The estimate of $\int_{0}^{1}\cos(x^{2})dx$ using the Trapezoidal Rule with $n = 4$ is $0.8955$ and it is an underestimate.