i Apply the trapezium rule to find the approximate value of ∫0^1 x^3 d x with a four equal inte

step 1 So in this question, we're asked to use the trapezoidal rule to solve this. We have change of x

I Apply the trapezium rule to find the approximate value of...

i Apply the trapezium rule to find the approximate value of $$ \int_{0}^{1} x^{3} \mathrm{~d} x $$ with a four equal intervals b eight equal intervals. ii Find the exact value of $\int_{0}^{1} x^{3} \mathrm{~d} x$. iii Determine the percentage error between the exact value and the estimated values found in $\mathrm{i}$. iv Comment upon your results to iii.

i Apply the trapezium rule to find the approximate value of
$$
\int_{0}^{1} x^{3} \mathrm{~d} x
$$
with a four equal intervals
b eight equal intervals.
ii Find the exact value of $\int_{0}^{1} x^{3} \mathrm{~d} x$.
iii Determine the percentage error between the exact value and the estimated values found in $\mathrm{i}$.
iv Comment upon your results to iii.

Key Concepts

Numerical Integration

Numerical integration involves approximating the definite integral of a function, especially when the integral cannot be computed analytically or when a quick estimation is needed. This method partitions the interval of integration into subintervals and sums contributions to produce an estimate of the whole area under the curve.

Trapezoidal Rule

The trapezoidal rule is a specific technique in numerical integration that approximates the area under a curve by dividing the interval into subintervals and approximating the function with straight-line segments over each interval. By representing each segment as a trapezium (a quadrilateral with parallel sides), the area is computed and summed, providing a practical estimation of the integral.

Subdivision of the Integration Interval

Dividing the integration interval into equal subintervals is a fundamental step in many numerical methods. The accuracy of these methods, including the trapezoidal rule, typically improves with the number of subintervals because smaller segments allow the approximation to more closely follow the curve of the function.

Exact Integration via Antiderivatives

Exact integration is the process of finding the precise value of a definite integral by determining the antiderivative of the function and applying the Fundamental Theorem of Calculus. This approach is used when a function has a known antiderivative, allowing for an exact analytical solution without resorting to numerical approximations.

Error Analysis and Percentage Error

Error analysis in numerical integration quantifies the difference between the estimated value and the exact value of an integral. Percentage error is a common metric, expressed as a percentage of the exact value, which helps assess the accuracy of the numerical method and guide improvements in the approximation process.

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