Math Problem Statement
Use a series to estimate the following integral's value with an error of magnitude less than
10 Superscript negative 510−5.
Integral from 0 to 0.4 sine x squared dx∫00.4sinx2 dx
Question content area bottom
Part 1
Integral from 0 to 0.4 sine x squared dx∫00.4sinx2 dxalmost equals≈enter your response here
(Do not round until the final answer. Then round to five decimal places as needed.)
Solution
To estimate the integral using a series with an error magnitude less than , let's proceed with the Taylor series expansion of around :
-
The Taylor series for is:
-
By substituting for , we get the Taylor series for :
-
To estimate , we integrate each term in the series expansion term by term:
-
Now, we integrate each term:
Since , we do not need additional terms to achieve the required accuracy.
-
Adding the results:
-
Rounding to five decimal places:
Thus, the estimated value of the integral to five decimal places is:
Would you like further clarification or more details on the series expansion?
Here are some follow-up questions to consider:
- How would the estimate change if we required an error of instead of ?
- Could we use a different series expansion method to improve convergence?
- What are the general rules for determining the number of terms needed in a Taylor series approximation?
- How would the answer change if the interval were from to instead of to ?
- Can we apply Simpson's Rule to estimate this integral with the same accuracy?
Tip: Using the Taylor series method is effective for small intervals where the series converges quickly.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Taylor Series Expansion
Integration
Error Bound in Series
Formulas
Taylor series for sin(x): sin(x) = x - x^3/3! + x^5/5! - ...
Taylor series for sin(x^2): sin(x^2) = x^2 - x^6/6 + x^10/120 - ...
Integral approximation using series expansion: ∫sin(x^2) dx ≈ ∫(x^2 - x^6/6 + x^10/120 - ...) dx
Theorems
Taylor Series Theorem
Error Bound in Taylor Series
Suitable Grade Level
College Calculus or Advanced High School (AP Calculus)
Related Recommendation
Error Estimation Using Trapezoidal Rule for sin(x^2) from 0 to 1
Approximating f(x) = sin(x) Using Taylor Series Near Zero
Simpson's Rule Approximation of sin(x) from 0 to π with Error Analysis
Integral of sin(x^5) - Approximation Using Series Expansion
Approximation of Integral Using Maclaurin Series for sin(x^2)