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

Ask by Donnelly Todd. in the United States

Mar 22,2025

Question

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

Ask by Donnelly Todd. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

We approximate the integral

$$
\int_{-3}^{3} e^{-x^2}\,dx
$$

using the trapezoidal rule with $$ n=6 $$ trapezoids.

1. Calculate the width of each subinterval:

$$
h = \frac{b-a}{n} = \frac{3 - (-3)}{6} = \frac{6}{6} = 1.
$$

2. The sample points are:

$$
x_0=-3,\quad x_1=-2,\quad x_2=-1,\quad x_3=0,\quad x_4=1,\quad x_5=2,\quad x_6=3.
$$

3. The trapezoidal rule formula gives:

$$
\int_{-3}^{3} e^{-x^2}\,dx \approx \frac{h}{2} \left[f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + 2f(x_5) + f(x_6)\right].
$$

4. Evaluate the function at the sample points:

$$
f(-3) = e^{-(-3)^2} = e^{-9},\quad f(-2) = e^{-4},\quad f(-1) = e^{-1},
$$
$$
f(0) = e^{0} = 1,\quad f(1) = e^{-1},\quad f(2) = e^{-4},\quad f(3) = e^{-9}.
$$

5. Plug these into the formula:

$$
\int_{-3}^{3} e^{-x^2}\,dx \approx \frac{1}{2} \left[e^{-9} + 2e^{-4} + 2e^{-1} + 2(1) + 2e^{-1} + 2e^{-4} + e^{-9}\right].
$$

6. Combine like terms:

$$
= \frac{1}{2} \left[2e^{-9} + 4e^{-4} + 4e^{-1} + 2\right].
$$

7. Factor out 2 from the expression inside:

$$
= \frac{1}{2} \cdot 2 \left[e^{-9} + 2e^{-4} + 2e^{-1} + 1\right] = e^{-9} + 2e^{-4} + 2e^{-1} + 1.
$$

8. Substitute approximate numerical values:

$$
e^{-9} \approx 0.000123,\quad e^{-4} \approx 0.018316,\quad e^{-1} \approx 0.367879.
$$

Then,

$$
e^{-9} + 2e^{-4} + 2e^{-1} + 1 \approx 0.000123 + 2(0.018316) + 2(0.367879) + 1.
$$

$$
\approx 0.000123 + 0.036632 + 0.735758 + 1.
$$

$$
\approx 1.772513.
$$

9. Rounding to the nearest thousandth:

$$
\approx 1.773.
$$

Thus, the trapezoidal rule approximation is

$$
\boxed{1.773}.
$$
The trapezoidal rule approximation of the integral $$ \int_{-3}^{3} e^{-x^{2}}\,dx $$ with $$ n=6 $$ trapezoids is approximately **1.773**.

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

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Calculus Questions

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

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Calculus Questions

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