Use the trapezoidal rule to
approximate the integral with
rapezoids. Round your answer
to the nearest thousandth.

Ask by Bond Ford. in the United States

Mar 22,2025

Question

Use the trapezoidal rule to approximate the integral with $$ n $$ rapezoids. Round your answer to the nearest thousandth. $$ \int_{0}^{1} \sin x^{2} \mathrm{dx} \approx[?] \quad \mathrm{n}=5 $$

UpStudy AI · Accepted Answer

We first note that the trapezoidal rule approximation for an integral

$$
\int_a^b f(x)\,dx
$$

with $$ n $$ subintervals is given by

$$
T_n = \frac{h}{2}\left[f(x_0) + f(x_n) + 2\sum_{i=1}^{n-1} f(x_i)\right],
$$

where

$$
h = \frac{b-a}{n},
$$

and $$ x_i = a + i\cdot h $$ for $$ i = 0, 1, 2, \ldots, n $$.

For the integral

$$
\int_{0}^{1} \sin(x^2)\,dx
$$

we have $$ a=0 $$, $$ b=1 $$, and $$ n=5 $$. Thus,

$$
h = \frac{1-0}{5} = 0.2.
$$

The nodes are

$$
\begin{array}{ll}
x_0 = 0, & x_1 = 0.2, \
x_2 = 0.4, & x_3 = 0.6, \
x_4 = 0.8, & x_5 = 1.0.
\end{array}
$$

We now evaluate the function $$ f(x)=\sin(x^2) $$ at these points:

$$
\begin{array}{ll}
f(x_0) = \sin(0^2) = \sin(0) = 0, \
f(x_1) = \sin(0.2^2) = \sin(0.04) \approx 0.040, \
f(x_2) = \sin(0.4^2) = \sin(0.16) \approx 0.159, \
f(x_3) = \sin(0.6^2) = \sin(0.36) \approx 0.352, \
f(x_4) = \sin(0.8^2) = \sin(0.64) \approx 0.598, \
f(x_5) = \sin(1^2) = \sin(1) \approx 0.841.
\end{array}
$$

Plugging these into the trapezoidal rule formula gives

$$
T_5 = \frac{0.2}{2}\left[f(0) + f(1) + 2\left(f(0.2) + f(0.4) + f(0.6) + f(0.8)\right)\right].
$$

Simplify:

$$
T_5 = 0.1\left[0 + 0.841 + 2(0.040 + 0.159 + 0.352 + 0.598)\right].
$$

First, sum the interior function values:

$$
0.040 + 0.159 + 0.352 + 0.598 = 1.149.
$$

Multiply by 2:

$$
2 \times 1.149 = 2.298.
$$

Now add the endpoints:

$$
0.841 + 2.298 = 3.139.
$$

Finally, multiply by 0.1:

$$
T_5 \approx 0.314.
$$

Thus, rounding to the nearest thousandth, we have

$$
\int_{0}^{1} \sin x^2\,dx \approx 0.314.
$$
The approximate value of the integral $$ \int_{0}^{1} \sin x^2 \, dx $$ using the trapezoidal rule with $$ n = 5 $$ is **0.314**.

Use the trapezoidal rule to
approximate the integral with
rapezoids. Round your answer
to the nearest thousandth.

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Use the trapezoidal rule to approximate the integral with rapezoids. Round your answer to the nearest thousandth.