The problem requires estimating the integral $\int_0^2 e^{-4x^2} \, dx$ using a Taylor polynomial of degree 4 centered at 0. Here's how to approach this:

Step 1: Taylor Polynomial for $e^x$

The Taylor series expansion of $e^x$ about $x = 0$ is: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$

Since we need to approximate $e^{-4x^2}$, we substitute $x = -4x^2$ into the series: $e^{-4x^2} = 1 - 4x^2 + \frac{(-4x^2)^2}{2!} - \frac{(-4x^2)^3}{3!} + \frac{(-4x^2)^4}{4!} + \cdots$

Simplify terms up to degree 4:

• $1$
• $-4x^2$
• $\frac{(-4x^2)^2}{2!} = \frac{16x^4}{2} = 8x^4$

Thus, the degree 4 approximation is: $T_4(x) = 1 - 4x^2 + 8x^4$

Step 2: Integral Approximation

Substitute $T_4(x)$ into the integral: $\int_0^2 e^{-4x^2} \, dx \approx \int_0^2 \left(1 - 4x^2 + 8x^4\right) \, dx$

Now, compute each term:

1. $\int_0^2 1 \, dx = [x]_0^2 = 2$
2. $\int_0^2 -4x^2 \, dx = -4 \int_0^2 x^2 \, dx = -4 \cdot \left[\frac{x^3}{3}\right]_0^2 = -4 \cdot \frac{8}{3} = -\frac{32}{3}$
3. $\int_0^2 8x^4 \, dx = 8 \int_0^2 x^4 \, dx = 8 \cdot \left[\frac{x^5}{5}\right]_0^2 = 8 \cdot \frac{32}{5} = \frac{256}{5}$

Combine terms: $\int_0^2 e^{-4x^2} \, dx \approx 2 - \frac{32}{3} + \frac{256}{5}$

Step 3: Simplify Result

Find a common denominator (15):

• $2 = \frac{30}{15}$
• $-\frac{32}{3} = -\frac{160}{15}$
• $\frac{256}{5} = \frac{768}{15}$

Add these: $\frac{30}{15} - \frac{160}{15} + \frac{768}{15} = \frac{638}{15}$

Thus, the approximation is: $\int_0^2 e^{-4x^2} \, dx \approx \frac{638}{15}$

Would you like a detailed explanation of any specific step?

Related Questions:

1. What is the general formula for Taylor expansions, and how do we apply them to composition of functions?
2. Why do we truncate the Taylor series at degree 4 instead of including higher terms?
3. How does the Taylor expansion help in approximating difficult integrals like this?
4. What is the role of the even symmetry of $e^{-4x^2}$ in this integral approximation?
5. How would increasing the Taylor polynomial degree affect the accuracy of the approximation?

Tip:

When approximating integrals using Taylor polynomials, always ensure you compute up to the required degree accurately and note which terms significantly contribute to the integral value.